I learned it the wrong way, but it works perfectly for me.
I use the nail on my middle finger to graze the string.
I don't even think about it anymore, my middle finger just kicks up when I want the harmonic to sound.

-Eddie

If you can get a copy of the Eagles "Farewell 1" tour video, there are some great shots of their guitarist doing this. I believe his name is Steuart Smith. He uses a thumb pick and reaches over with his middle finger to make the harmonic. I use the rounded edge of the pick all the time a la Robben Ford and Larry Carlton, and I just choke up on the pick and I can usually do the harmonics pretty easily - on Strat or any other guitar.

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Frank
A legend in my own mind.

I just use light pressure, (not pushing the string all the way down to the fret) from my string finger and pull off *just as I pick the string* with my pick. Timing is crucial, more so than the pick or how I hold the pick.

Just my 2 cents.

Gridlok

I agree that a light touch is key and timing is crucial. The touches have to be simultaneous or you'll just the original note ringing or you'll get a dull "plunk".

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Frank
A legend in my own mind.

Well the easiest way to do this is to choke up on the pick where just a tiny bit is sticking out and remember to dig into the note and pick down. Also different harmonics are sounded as all ready mentioned by moving your picking to different spots. Take the G string and pick up and down the string and you will find spots where they just jump out.

Another cool trick is to sound a natural harmonic at the fifth or seventh fret on the G string and then bend the string behind the nut for that outer space sound.