Great advice from Steve, Jack, Bit, and RT; I couldn't agree more.

In many products, the rocker power switch is simply a single-pole, single-throw interrupt switch between the power cord and the first electrical component connected to the black wire (hot lead) inside the chassis. Electrically, there is almost zero difference between throwing this switch and throwing a switch on a power strip upstream. If anything, the power strip method would be even more gentle because you have the resistance and capacitance of the power cord to help buffer any spikes that are created.

If there is no way to turn a product off aside from unplugging it, I think a power strip would be preferable to yanking the power cord, because you leave the neutral wire (and ground, in the case of a three-prong device) connected, which seems less disruptive than pulling them all at the same time. The ground prong, in particular, is used for safety and as a ground reference, so keeping that intact seems like a good idea.

I do it....

I wired a bunch of 4x4 boxes with duplex outlets..(normal home style outlets) and had them all powered on and off by a switch.... I knew I was planning a studio and lots of gear that needed to be turned on & off.

The only exception is of course, the computer. Using a surge suppressor bar is a really good idea... since there are always surges coming in to the equipment by way of the power line.....

yeah... use surge suppressors.... good ones.

I had used power strip for many years and haven't found any power issues, though I'm not sure whether it is safe. I just change another power strip , it is a desktop one from bestek. I love it, not because it has well-organized multiple outlets and superior surge protection, but also the bestek desktop power strip has independent switches to let you control the each AC outlets easily. There also some other power strips that each outlet is controllable by a single switch. That is a good way to save time and energy. If you have such concern, you can try a power strip with independent switches. 

A quick and easy way to see if using your power strip will cause damage to your equipment by turning everything on at once is to hook up a digital volt and amp meters to the outlet the power strip is plugged into to measure and get visual feed back of the draw and surge caused by powering up multiple pieces of gear at once.
 If your voltage dips and or amperage spikes above the power strip's rating upon power up, even though the PS's circuit braker may not pop, is hard on all the equipment plugged into it as each piece struggles to suck up the necessary supply of power it needs thru a "straw too small to handle the shock instantaneous volume of wattage" required to power up everything at once 
 Hear any pops, clicks, or crackles during power up or shutting down? That's the sound of virtually kicking capacitors square in their ball sacks, and while they can typically handle years of this, they still do in fact ever so slowly die a premature death while slowly, but surely ever so subtly building up, adding noise, distortion, and slowly but absolutely degrading over all sound quality over time.
 It typically builds up and happens so slowly over time it's hardly noticeable until it is, at which point is typically too late as damage cause by weakened and leaky capacitors takes their toll on other circuit board components (slowly being starved to death) and struggling internal component power supplies.
 Not to even mention the damage it does to your "Mojo" or "groove" which has a bad effect on the over all quality of your work and full dead on accuracy mixing potential, as in....
 Accurately and critically getting "Out" of your tracking sessions as what you put "In" to them..
 And THAT my fellow CakeHeads, is why after so many decades of earnestly trying my best to perfect my crafts and skills, I settled happily and firmly confidently into SONAR Platinum and Focusrite.
 Because as far as I'm concerned, when I'm recording and mixing music "Sound is Everything" and to me my studio is a very high performance vehicle that I must make sure has enough gas and octane to get me from start to finish with little to NO issues with maximum efficiency.
 You wouldn't expect your car to run good on dirty gas, why would you expect your recording rig to run good on dirty electrical power?
 
 If you can't seem to fully grasp of understand how important this, just ask any guitarist who just had his vintage Fender amp recapped, or even someone who replaced the caps in the built in power amp of their near field studio monitors who are typically nothing short of SHOCKED by the difference of noticing the over all bump in clarity or what they've been missing!
 Nor does it take a whole lot of critical listening to realize how much noise you "DON'T" hear any more by replacing a cheap power strip with a prograde power conditioner, and or notice a difference in performance and reliability of each piece of equipment you plug into it.
 
 Living and recording in this rather charming 80+ year old house mostly wired with old wiring, I'm a HUGE fan of power conditioning power strips for studio, amp, and instrument equipment, all running off of any one of 3 Furman power conditioning units that I power up sequentially a few seconds at a time, and 2 APC UPS (uninterrupted power supplies.) for computer/workstation/ network, printer, and such.
Not only do they protect the equipment from surges and spikes, they isolate and defeat hum and stray RF noise (radio freqs) transmitted thru cheap and dirty A/C power that flows freely thru cheap power strips, even from one amp or unit to the next and or thru the many cables and connectors hooked up in the studio.
 
 Can't locate that occasional intermittent hum that's been driving you crazy for God only knows how long?
 Well you could (may) identify and isolate it by unplugging the refrigerator, or the microwave oven in the kitchen, the heating and or air conditioning unit(s), or perhaps it's just a dimmer switch used for lighting or ceiling fan or window fan anywhere in any room of the house or apartment your in.
 Or get power conditioning and a UPS and forget about these annoyances even happen and or ever existed, and have a nice day!

I have everything going to a Furman line conditioner from there a few other basic power bars mounted on the wall behind things and on on the desk to use when needed. 
And my Power amp has a delay so that solves that but for sure first thing I do before powering down is turn the master volume on my interface off. 
Of course the computer is last on and first off using the power button for on and OS for off. 

There is a simple solution.  Use a power board with switches for each outlet.  I have my computers running on 24/7 power independent of this  (but still coming from a master computer power switch so that I can turn them all off if I need to) 
 
I have got my entire studio on 4 circuits on the power board.  Switch them on sequentially.  One at a time.  That way you avoid any power surges.  I have never had an issue doing it this way.  Your equipment will not mind having their power switches permanently on and being remotely switched either.  If you are smart you can arrange the 4 circuits in such a way as that if you only need to do certain jobs then you only need to switch on one or two circuits.  For a full job then in my case I need all four.
 
The way I have set up the 4 circuits is:
 
1 All lights 
2 Basic setup. e.g. mixer, general power circuits that need to be on in order to hear audio. My audio interface is inside my computer but if yours is external this is the circuit it should go to. 
3 Synth rack and keyboards
4 Powered monitors
 
So if I only need to edit some audio (in the daytime) then I only need circuits 2 and 4.  On headphones circuit 2 only. 
 
Zone your studio into power zones according to the types of jobs you do.  That way you can leave some circuits off and save power if you are only doing certain things.
 
Having the powered monitors on my circuit 4 also works well because I can switch on the studio on the order 1 to 4 but turning off I go 4 to 1. e.g. monitors off first. 
 
From the switching power board obviously you will have other power boards connected to each circuit in order to be able to feed the rest of your gear.
 
When I go away I switch everything off  (computers as well)  but also remove the power plug from the wall to be on the safe side.  My whole studio is connected to one wall socket.  You avoid earth loops this way. e.g. with everything I own in the studio on, the total power draw in my case is 800 watts. 
 
It is not a bad idea to get a hold of a power meter  (I had an electrician next door so I used his)  and measure how much power each section draws.  It can be quite revealing.  It is also worth going right through your whole house and test every appliance as well.  That is also very revealing. e.g. my coffee machine draws 1000 watts when it is heating up etc.  I used to leave this on a lot but don't now!

there are also some power distribution units that can be sequenced - so you can fire up your devices in order to avoid pops etc. some of my power amps have that option as well. Jeff Evans' approach is basically a manual approach which is similar to how I manage the power up and down options. 

Like JackN2 says, the power strip switches will likely bounce which is bad, but using higher up Furman or other quality strips for master on/off not bad.

I have been using power strip switches for 30 years and they have never gone bad so I say they are fine. Just make sure the strip is decent quality and the switches are as well.

This quote from above is key "Just make sure the strip is decent quality and the switches are as well."
Which means the obvious, the good ones aren't bad....hehehe.
The cheapest also have MOVs across L1, Gnd, Neutral; so the mfg can say technically "with lightning and surge protection with xxx joules. These can be super cheap and cause power hum as they age and the surfaces of the inside of the discs become pitted and there is a tiny bit of continuous contact between the two discs. *Which are supposed to arc over evenly in an over voltage situation.
Here again the better, like Furman, others, use better of this MOV and SAD technology for TVSS protection, and if any failures trip the small ckt brkr on it. Cheap ones, like 1.99 strips, have caught on fire in the past.

I wouldn't imagine anyone here investing in audio interfaces then use a 1.99 item. There are many cases of power hum being eliminated or greatly reduced by using quality power distribution.

Like anything, common sense prevails.