Over the years, mine has become smeared with an assortment of grime and fingerprints. Is there a safe cleaning method?

They sell cleaning pads.

Is it glass? Then clean it with some watered down windex.

If it's an lcd, then mix isopropyl alcohol (rubbing alcohol) with water (roughly 50-50... but I usually use about 30% alcohol and 70% water) and clean it with a soft towel. Be sure not to press hard.

Thanks for the heads up. I'll try the 30%-70% solution on an old monitor and post back.