As above, you need to capture the "warmth" or clarity or whatnot during, well, the capture stage.  You can fix off notes post, too, but it ain't the best method, esp. if you need to pitch correct the whole damn thing.
 
I find that input transformers make all the difference, and good input equipment.  I just had the good fortune to put the new WARM Audio LA-2A through some paces, and it is a really nice piece of kit.  It adds saturation as it is driven, and more saturation when driven more, little hairs all over the recording that just tickle my audio fancy.  But not distortion, nothing harsh, just input sounding bigger and bigger with that transformer thickening I associate with warmth.
 
It is a lot easier than trying to recreate warmth after the fact.  Or I've found that so, although the CA-2A is nice, as is the SSL bus comp.  Both those can thicken things up a little, just as the 1176 can put an edge on a recording.  But inputting through good transformers is easier and a lot more fun when you get into the into the recording moment.
 

I agree with Eik.The pro channel consoles work well at giving you that "undefined and yet there" softness. I play a lot of contemporary jazz, the conventional wisdom being, the cleaner, the better. Recently, a song of mine was buggin' me because it was sterile and I found the console emulation on all tracks plus the tape module on the master bus still set to pretty clean gave the mix just enough "smear". Need more? Try the softube or tube saturation set for a subtle clip. One caution with the tube sat. emulations. Pick the tracks that need it, not just go global.My particular mix sounded great with just a little on the bass and vocals ( especially, the background)

Oh yeah, like you said, you can undo it too. Nothin' makes my teeth grind more than getting a perfect performance that is too processed, or too hot, thus making it a great unusable take.
 

Reckedtrek
Chevy, at the risk of being painfully simplistic, you said 
 
"I admit approaching the recording aspect from getting the clearest, not the warmest, sound I can. I'm thinking you can warm up a track well enough with the right tools, but you can't clean it up if need be later on.)  And whatever processing I apply to the master mix doesn't seem to have the goodies my ear desires. "
 
The two bold statements make me wonder why you don't experiment with getting warmer sounding tracks to begin with? I understand you're point about not being able to "undo warmth" but if that's the sound you're after maybe it's just question of finding how much warmth to allow on the tracks themselves. Maybe I'm off mark or being overly simplistic, but just thought I'd throw it out there. Hope you find what you're looking for one way or another! 

Yes, certainly did occur to me to just record them warm in the first place, but like a certain really clean guitar part, you can add an amp sim or pedal or whatever and "rock" it up a bit later on... doesn't always work the way I'd like...   But to add that special warmth to an entire mix without it sounding cheezy or something... I dunno, just the way I've approached things so far...  and believe me, I can have it more backwards than anyone. Lots to learn yet! This just made sense, trying different recording takes and picking what seemed the best at the time, and then as I progressed into the next song to record / mix, things may be rebalancing in another direction. 

AT
As above, you need to capture the "warmth" or clarity or whatnot during, well, the capture stage.  You can fix off notes post, too, but it ain't the best method, esp. if you need to pitch correct the whole damn thing.
 
I find that input transformers make all the difference, and good input equipment.  I just had the good fortune to put the new WARM Audio LA-2A through some paces, and it is a really nice piece of kit.  It adds saturation as it is driven, and more saturation when driven more, little hairs all over the recording that just tickle my audio fancy.  But not distortion, nothing harsh, just input sounding bigger and bigger with that transformer thickening I associate with warmth.
 
It is a lot easier than trying to recreate warmth after the fact.  Or I've found that so, although the CA-2A is nice, as is the SSL bus comp.  Both those can thicken things up a little, just as the 1176 can put an edge on a recording.  But inputting through good transformers is easier and a lot more fun when you get into the into the recording moment.
 

Ahh...     the WARM LA-2A and the transformer thing seem like a great way (or maybe the ideal way) to go from your description...  warmth, saturation, thickness, without actually hearing specific distortion artifacts. The sort of tone / feel I love as a guitar player.  
Alas, outboard gear in general is way, way out of my snack bracket. 

I fully understand the OPs point of view.
At least in my case, with less than perfect monitoring and not so analytic ears, if I try to make the sounds warm from the beginning, I end up with something foggy, muddy, that lacks sparkle. From there, it's quite hard to move towards
the sparkly side.
If I try to create a clear, distinct soundscape with sparkle, I can soften the clarity and spark little by little.
 

I've gone from recording guitars and bass mic'd off the cabs, to full clean guitars/bass, re-amp'd with sims, to outboard sims direct to track, to a bit of everything, and now, completely back to mic'd cabs with no 'extra' clean signal double tracked. Same with acoustic, straight off the mic's only, no piezo clean out double tracked. This way, I usually don't have enough 'highs' to worry about cutting, accept maybe the acoustic stuff.
 
I've find the 'clean' approach doesn't work at the master mix stage, just like you. Without the extra 'noise', the final result is just too squeaky clean and unbelievable. And I'm doing country/pop, not rock.
 
For drums, it's compression, and reducing highs particularly for the kick and snare. And anything with a good rock type kit, parallel, heavy compression, is a must for me. I use the saturation knob and the tube warmer in the PC a lot, and console emulation on almost everything.
 
The only non-cake stuff I use is IK's T-racks, usually for the vintage gear, and I'm about to start messing with Ozone.
 
Some of this could come down to your room/monitors. If you could find a way to accentuate the mid/highs and highs, it might make you mix a little darker.

Chevy
Trying to come up with a nice warm tone for some of my rock-ish master mixes. I've just compared a few of my tracks tonally to relatively recent tracks from bands like Aerosmith and Bon Jovi, and those tracks most certainly (to my ears) sound like rock music. Du'h. They are warm, and there is a certain amount of distortion or saturation to them that is warm but not clearly identifiable as specific distortion...  it's just very smooth and sounds sorta like a live band would.
My master mixes, before processing, are too clear and sterile in comparison...  (I admit approaching the recording aspect from getting the clearest, not the warmest, sound I can. I'm thinking you can warm up a track well enough with the right tools, but you can't clean it up if need be later on.)  And whatever processing I apply to the master mix doesn't seem to have the goodies my ear desires. I don't have a good Kramer Tape type plugin, apparently something like that would help...  wondering how you might recommend I get that warmth and the "friendly" distortion in a predictable fashion. 

How are guitars being recorded? If direct, then you're going to need some great amp sims, but if recorded with a mic, here's a few suggestions I have:

1) Ideally, if the guitars are recorded well, they won't need anything done to them, but this is rarely ever the case. But, with that said, I nearly always use a decent tube mic directly in front of a speaker cone along with an SM57, or something like that when tracking, carefully eliminating phase issues.

2) After recording, on the track itself, I normally will drop in Waves NLS Spike, Channel 29, drop off the Noise and Mic buttons, and goose the Gain a bit. You can do this similarly in Platinum's S-Type Console Emulation, although they differ slightly. Something about Channel 29 in the Spike NLS...
 
3) EQ out the very deepest lows and extreme highs. Leaving these in will needlessly over-work any compressors or limiters you may use. However, leave in a little of the 100-200Hz range. You'd be surprised at how this will thicken up the guitar mix, and doesn't really over-shadow the bass as much as some might expect.

4) At this point, I use 'modeled' EQ's, such as Waves SSL E-Channel or the G-EQ, API-550, Kramer HLS to add some 250-500Hz boost (slight) to add some modeled warmth. The HLS does this really well. However, this is the fun of it all. TRY THEM ALL! :)
 
5) Don't over-compress or limit. Some of the warmth comes from edginess.
 
6) Create a Send to a Reverb buss and put a Convolution reverb on that buss, such as Waves L1, Altiverb 7, or EastWest Q-Spaces. They will add some warmth on their own, if you've matched the right impulse settings (chose the right room space for the reverb). Don't overuse this, though, and mix it in slowly. 

7) Bypass often to see how much improvement you've made, or if something's gotten worse.

Kalle Rantaaho
I fully understand the OPs point of view.
At least in my case, with less than perfect monitoring and not so analytic ears, if I try to make the sounds warm from the beginning, I end up with something foggy, muddy, that lacks sparkle. From there, it's quite hard to move towards
the sparkly side.
If I try to create a clear, distinct soundscape with sparkle, I can soften the clarity and spark little by little.
 

Ah, yes, well put..., my thoughts just in better words. Yeah, if the original (say guitar) tracks are warm, they just don't seem to have what it takes to stand out in the mix;  with an overall muddy effect. But I am still missing that over-all mix warmth (distortion I'll call it) that adds that certain magic to clean tracks. The earlier response about input transformers saturating has me intrigued and interested.