I use Murphy's Oil soap and gently scrub dirty checkering with an old toothbrush, and then again with plain water to rinse. Blot dry rather than rubbing and risking breaking any diamonds. On really cruddy checkering, I use a solution of household ammonia which will saponify, or turn oils into a soap. Since soaps are water soluble, the oily grimy crud rinses away better. It shouldn't harm the wood any more than a morning of hunting in the rain. As you proceed to harsher chemical solvents, you run the risk of removing the finish.
