Is there a simple way to make a MIDI track not seem so mechanical?

As a rule, if all the notes are the same volume, then the off-beats will sound subjectively louder than the onbeats. So adjust accordingly. This seems to apply to halfs, quarters, eighths, sixteenths, or any length.

Frank's Midi Plug-Ins - the humanize one.
http://www.midi-plugins.de/
 

If it's for drums, and you use AD2, you can record the rhythm (or drag it, but it's a bit glitchy and I'm currently speaking about the drag function with XLN, who seem completly confused you can do it) into beats section. From their you have the 'Transform' tools. Allowing you to adjust the accents, and introduce randomness to the velocities and timings. When you are happy, drag it out to the clip

Good point, I almost never quantize to 100 percent. If I have to quantize at all I like to use 80 percent or maybe 85 tops.

Interesting article. You can't really blame the "android" effect of drum machines on any one factor like timing. The "human" feel isn't just one factor but a series of factors including sample used, velocity variances, note timing, etc. If a DM uses the exact same sample @ the exact same velocity level @ 100% quantization, you'll get the android effect.
 
Some better VIs provide these variances. Most run of the mill VIs don't, so it's up to the user to make them.
 
Another factor is offsetting each instrument so that your ears aren't hearing every instrument strike at precisely the same time.
 
Most, if not all, analog recordings, especially those on tape, drift throughout the entire mix, usually going from slow to fast. I think this "effect" is probably the most humanizing of all and the hardest to emulate digitally.

I read the article Craig linked and it is really good.  I am stupid, though, in that I get confused when they talk about moving events forward or back.  I understand early and late, but with forward and back, I am not sure which is which.

For MIDI programming to result in music that has intention, gesture, phrasing, dynamics an expression you can play the music in live or you can program.  If you program:
 
1.  Think like a player, meaning strong and weak beats.  Make some notes long, some shorter, some notes louder, some softer, some with fast attacks, some with slower attacks.  It all depends upon the phrase and the timbre you're working with.  Think in terms of natural phrasing, with very slight pauses, cadences, high and low melodic points and interesting harmonic rhythm. 
 
2. Use a lot of program/articulation changes, the more the better.
 
3.  Sometimes you might have to move an entire track forward or backward by a given number of ticks, to get a groove.  I find with softsynths and strings for example, the crisp attacks of softsynth timbres require me to move the entire softsynth track 15-25 ticks late.  This creates a groove that doesn't exist if I don't do that.  Here's an example: www.jerrygerber.com/mp3/Raga.mp3
 
4.  The quality of composition enters into the equation.  Sometimes a computer performance sounds mechanical not only because of MIDI performance issues, but because of poor voice-leading, unbalanced harmonies, wrong tempi, etc.  Some music requires a lot of tempo changes, use them, even very small changes.
 
5.  If a note sounds too early, move it later by a few ticks.  If it sounds late, move it earlier by a few ticks.
 
Jerry
www.jerrygerber.com

cpkoch
Unfortunately I am not an instrumentalist of any sort  so it remains impossible for me to record stuff from scratch as many have suggested in answer to my query.  Nevertheless, I certainly want to thank all of you for your wonderful suggestions.  I had no idea that my query would attract so much great attention!  Thanks!!

Don't let that stop you! Do you at least have a MIDI keyboard controller? You might be surprised at what you can do with two fingers and some practice. My granddaughter can play Jingle Bells with no training beyond Grandpa's encouragement. With input quantization turned on (at < 100%), I could probably record her playing a decent rendition. If you don't have a keyboard, they start as little as fifty bucks.
 
Can you sing? Audio-to-MIDI conversion can let you sing any part, convert it to MIDI and drive any instrument with it.

On feature in Sonar is Groove quantise. Rather than quantise to the grid it will use a pre defined groove or anything in your clipboard. This is more effective than pure random humanization.

Glyn Barnes
On feature in Sonar is Groove quantise. Rather than quantise to the grid it will use a pre defined groove or anything in your clipboard. This is more effective than pure random humanization.

I’m more than a bit surprised that it took so many answers before anybody mentioned Groove Quantise. As Graig mentioned, just adding random variations to a midi track (as many of these ”humanise” functions do) doesn’t make it sound like a musician actually played the thing. No player worth their salt makes random variations – all the variations in the timing and volume have a purpose.

I have found a few pops, the wife complaining about money, the kids arguing.. helps me humanize my tracks on a totally different level