Depends on the size of the part, but for small stuff I use thin brass wire with the end formed into a loop with a pair of really fine round nose pliers. Wire of between .010 and .020 seems to work the best for me. When it gets clogged, I just toss it and cut another piece off the spool (or the rod, for the thicker ones).
For flat parts, I dip the part into a small pool of liquid CA, bring it over to a nearby paper towel, wick off the excess and rapidly place the part in place. Not much room for error with this technique, but with practice it is precise and neat.
To hold parts, I have two really good pairs of tweezers that I really do my best not to abuse. Still, they need some work with a handstone from time to time to keep in top condition. When sharpened they will pick up a single hair with no trouble.
I also use a very sharp scalpel to move and postion parts that will not show the tiny pinprice it leaves. What is nice about that technique is that it holds pretty firmly, but releases well.
Kurt