I've had checkering clean up really well with a toothbrush and white cloth using lemon grass oil. I think this stuff is about as gentle as it gets. You can find it at hardware/home improvement stores. Use two white shopcloths, one to apply, a toothbrush (a worn out one discarded by the wife works best) and a clean, dry cloth to remove the dirty lemon oil.
You can use it on the whole stock as well. Takes plenty of old goop off, doesn't harm any remaining finish.
It won't totally clean up really nasty, skin oil/blood/grime soaked checkering, but, it works well, is cheap, and a great first step. If it doesn't clean up this way, you likely need a refinish, anyway.
Best,
Ted
