I have a few frequently played card games where the cards no longer feel "clean". They aren't stained or anything, just a bit grimy. Is there an effective way to clean the cards without damaging the graphics?
I've found that a slightly damp washcloth with a little bit of plain old bar soap on it works just fine -- don't use those goofy hand soaps with the beads or anything fancy in them.
Lightly rub the cloth on the card, dry it off with another washcloth, and let it sit for a few hours. You definitely want to try this on one card first to make sure that nothing streaks on the card, and only do small batches at a time to make it manageable.
I wouldn't do this to any Magic cards that you find valuable; those should have been in card sleeves in the first place. It might be safe, but I would hate to be responsible for damage to your Mox or Lotus.
Another option is to get a razor blade or hobby knife, and carefully scrape the grime off. You want to use a very light touch here to avoid scratching the card. Basically put the whole of the blade in contact w/ the card, just slightly off of vertical, and lightly scrape.
Regardless of what method you use, if the cards are used frequently prevention is the best method, but if you do have success cleaning cards, you'll definitely want to get some sleeves to keep your cards nice and clean.