Break your chain, then put it back together using a master link. When you want to clean the chain, remove it from the bike (takes about 10 seconds), put it in an old spaghetti sauce bottle (the Prego Italian Sausage and Garlic flavor bottles work the best, or so I've heard) about 3/4 full of mineral spirits. Shake vigorously for a minute or so. Come back 10 or 15 minutes later if you really want to and shake it again. Remove chain from bottle, hang up to dry for a short time. Reinstall on bike. Re-lube. Done. It's so much easier, and more thorough, than any wiping down with a rag can possibly be. I personally take the extra step of rinsing the chain off in a second bottle of cleaner mineral spirits.