Okay. I know about this b/c I did get a used monitor and it was pretty dirty when I got it.
First, get a good antistatic cloth. Spray a corner with either Windex or lcd/touch screen monitor cleaner (I like
Iklear stuff because it does the job and I have it around for my Ipod Touch anyway) and wipe said corner all over the screen. Do NOT directly spray the screen; my married sister's little 4 year old son did that with Windex on a LCD monitor, and it destroyed the monitor. You don't want that.
Then just dry it/polish it with the dry part of the cloth. It should be clean after that.