Sometimes, soap and water just won't cut the grime that's on the box. That's when I pull out Orange Glo. It's an all-natural cleaner made specifically for wood. However, it works wonders on stubborn grime and really makes the chrome and antennas shine on these old boomboxes. The trick is not to use too much. All you need to do is lightly mist a cleaning cloth with it.
If you're close to Canada, you can look for Almond Glo. It will do the same thing.