Are you asking advice on good cleaning products, or on how to keep the cabinets from getting dirty again after being cleaned?
If the former, you need to tell us whether they're painted, stained, or ??, and what are they made of?
If the latter, you can clean off the "dark and gummy" but obviously it will come back eventually. Why not add handles, knobs, or some other way to open the cabinets?

I like using liquid gold to clean the surfaces along with Murphy's Oil Soap. I agree with rpxlpx that you should look into adding pulls or knobs.