Apple offers a simple three-step guide for cleaning your LCD monitor:
1. Turn off the computer or display. You may need to turn off the computer in order to turn off some Apple displays.
2. Dampen a clean, soft, lint-free cloth or paper with water only.
3. Wipe the screen. Do not spray liquid directly on the screen.
This may have been common sense to many of you, but I'd been putting off a much-needed clean for a while because I didn't want to destroy the beautiful, mysterious screen. Since my instinct was to use some rubbing alcohol or glass cleaner (boy was I off!) to get the job done, I'm very glad I checked beforehand. Alcohol on your monitor can cause damage to the coating and dry out and crack the plastic. You can use mild, non-alcoholic detergents (and Apple does endorse one), but I'll probably stick with water for now.