Since many cabinets are not finished on the interior, you need to either have these areas finished and sealed or settle for wiping them out with a dry cloth. If you have access to running water and bleach you've got everything you need to get started on the road to cabinet cleanliness. Our first quick fix involves a rag or towel dampened with a solution of 3/4 cup of bleach per one bucket of water. Once applied, carefully wipe all cabinet surfaces. Next, go over everything again with a plain, wet cloth. Throughout the process, make sure to protect your skin from the harsh properties of bleach by sporting rubber gloves. Also, make sure to wear clothes you really don't care about, in case a few errant splashes of bleach come into contact with your duds.
If you don't have the time or inclination to mess with mixing things together, you can pick up a container of standard disinfecting wipes. Generally, these ready-to-use wipes are approved for use on polyurethane treated wood surfaces, but check your specific brand to be certain.
When grease is the problem -- as is often the case with lower cabinets under stoves -- a standard degreasing agent is generally all it takes to get things spic and span again. Fantastik or 409 are good products to start with.
Many people prefer to steer clear of chemicals as much as possible. For a green cleaning option, mix hot water with equal parts white vinegar. Dampen a rag and scrub gently.
No matter which cleaning method you choose, it's generally a good idea to run a dry cloth over the cabinet surfaces once they've been disinfected to keep them from streaking. Otherwise, restore the shine to your cabinetry with an approved polish and a dry rag.