Well the best way is to completely disassemble the radio and scrub the housing parts in a sink and rinse with hot water, then dry. If you're not willing to do that you can still spiff up a grungy radio.
First off, don't use an ammonia based cleaner on plastic. It can leave a haze when it drys.
For a quick clean up I use Simple Green and an old toothbrush. Lightly spritz the bristles and then work that into all the cracks and crannies of the radio. Wipe with a towel and then use compressed air (canned air works good) to blow out the last of the crud/cleaner and wipe again.
The idea is not to get the radio sopping wet. You want just enough cleaner on the bristles to dampen a spot and foam up as you scrub. Also letting the cleaner dry is counter productive, so doing section at a time is the best way.
Anybody else with suggestions?