Recently someone asked me if it was okay to spray Windex on her monitor. I think my shriek of horror startled her. It is NOT okay to spray Windex on a monitor, nor anything else. In fact, when it comes to cleaning an LCD screen, there's a right way and a wrong way. Let's focus on the right way (natch).
For starters, turn your monitor off. In fact, if you want to eliminate any risk of shock or other electrical damage, unplug it altogether. (I really don't think this is necessary, but better safe than sorry.) There will be some moisture involved in this cleaning, and the last thing you want is for liquid to come into contact with anything powered.
Next, find a clean, soft cloth (paper towel will do, but it's far from the best choice), then dampen it with water. Don't soak it--there should be no dripping to speak of--just get it moist. And don't use anything other than water. Ammonia- and alcohol-based cleaners can ruin an LCD. If your screen is seriously gunked up, and plain water doesn't get the job done, here's a homebrow option: create a 50/50 solution of distilled water and white vinegar.
If you decide to use a spray bottle, make sure you don't spray it directly on the screen. That's because drops can seep in around the bezel (and/or land in the keyboard), and, again, moisture and electronics don't mix. What you can do is spray your water or solution onto the cloth, then wipe the screen. Use a circular motion, which is best for eliminating streaks, and apply only light pressure.
Finally, let the screen air-dry for a few minutes, or gently wipe it dry with a fresh cloth. Only when you're absolutely certain it's 100 percent dry, plug it back in and turn it on. And there you go: good as new!
Now keep those smudgy fingers of yours off that precious LCD!