Summarize:

Empty out the contents and set them aside. Wash the insides of the cabinet using the soap solution and a sponge, and dry off the soap and water with a towel. Place some oil soap in a container and dilute it with warm water. Then, test the oil soap on an inconspicuous part of the cabinet's surface to make sure that it does not harm the finish. The sponge should have both a soft side and a scouring side for best results. If you have a layer of grease or other material that is stuck to the surface, then scrub the material with the scouring side of your sponge.  Doing this will remove any moisture or residue. Oil soap will eliminate your wax finish, so you should apply furniture wax or furniture polish to restore the cabinets' shine.
Open the cabinet doors. Mix some mild soap and water together in a small bucket. Close the cabinet doors. Dip a sponge into the oil soap. Wipe the surface of the cabinets with the sponge. Rinse your sponge frequently so that you don't reapply dirt that you've already removed. Dry the cabinets with a clean, lint-free cloth after cleaning them. Reapply wax finish if desired. Return the contents of your cabinets to their original location.