Well christ, if we're going to do it, we might as well do it right...
Most people here use detergents rather than solvents for chains. Solvents are stuff that just plain dissolves grease and lubes (mineral spirits, MEK, diesel, alcohol to some extent), while detergents are chemicals that allow grease and lube to become water soluble (simple green, almost all bike-specific degreasers). I'm of the opposite opinion, I won't put detergent into anything I'm not 100% sure I can get every last bit of it out of, since detergent residue will just sabotage whatever lube you put in after cleaning. Chains have tiny little gaps and such that I just really don't believe can ever be truly flushed of detergent. I also don't find that detergents work very well against bike grease, you tend to end up with a tacky film on the parts which is the heaviest oils and borderline varnish that the detergent won't cut, whereas with a solvent what you end up with is some pretty stinkin' clean metal. I also don't believe in using a toothbrush or anything like that, a swipe with a paper towel gets rid of as much macrogunk as I care to manhandle, I'm happy to let chemicals and gravity do the rest. So here's what I do:
I keep a big glass jar mostly full of mineral spirits around the shop; this does take a pretty big jar, we're talking pasta sauce ones don't always work but do use something with a screw-on lid rather than a yogurt type container, those lids are not airtight. When a chain is dirty I pop off the master link and throw it in there. I keep the master link separate and clean it by hand, you can get to all the parts of a master link so elbow grease and maybe a few drops of lube are all that is needed. You can throw it in if you want, I just hate fishing them out. It hangs out in there anywhere from an hour to a few days before I ride the bike again, and every time I see it, I shake it. When I take it out, I lay it on newspaper, outside and in the sun if possible, and give it at least an hour to dry. These chains are clean after this treatment, there is no residue and no gritty sound if you twist them. Reinstall, relube, wipe off excess lube, and go. I get about 3-4 chains out of a jar before it starts getting really gross, at which point I let it settle for a day, decant through a coffee filter into a new jar, and chuck the old jar with a little bit of mineral spirits and a lot of gunk.