If you consider what kind of dirt gets on to kitchen cabinets, that should give you some ideas. Grease, skin oil, and food are the primary villians. Starting out with a mild solution of Dawn (or your favorite, Dawn worked best on the Exxon Valdez spill, better than commercial degreasers!) and hot water, with a sponge. You do not want to saturate the wood, so don't soak it, and don't let water sit on it for very long before you wipe it dry.
Whatever cleaning product you use, try it on part of the finish where you won't ever see it if the product damages it. This could be on the inside of a cabinet door, or somewhere else. If you have sticky residue from contact paper or the like, Goof Off works really well, but again, wear rubber gloves, and get lots of ventilation. No smoking.
Take your time, and you should end up with good results. If you have more questions, be as painfully detailed and specific as possible and we will do our best to help!