Fill a bucket or your sink with warm water. Add 1 squirt of mild liquid dish detergent. Stir to combine.
Dip a clean, soft cloth into the soapy water. Squeeze the cloth to remove excess water, and keep the cloth wet but not drenched.
Rub the caked-on dirt with the cloth. Use firm pressure to work to release the dirt. Rinse the rag in the soapy water as it dries out. Squeeze out the excess water again, and continue to scrub until the dirt is gone.
Rinse the cloth under running water, and squeeze out the excess liquid. Use the wet cloth to wipe away the soap from the cabinet surface.
Dry the cabinet surface with a clean, dry cloth. Do not leave any water on the surface since water can damage wood.
Wet a white pad with water, and wring the excess water out of the white pad. You want it damp but not soaked. White pads are sponges with a white nylon scrubbing pad attached to one side. The white nylon pads are gentler than other scouring pads. Although you can use nylon pads of different colors, the white ones are free of dyes that can stain porous cabinet surfaces.
Sprinkle the damp white pad with baking soda. Apply enough to lightly cover the white pad's surface.
Scrub the surface of the cabinets with the pad. Use a circular motion, applying firm, even pressure to areas with heavy caked-on dirt. Rinse the white pad under running water every two to three minutes or as necessary. Wring the excess water out again, and apply more baking soda. Continue scrubbing the surface of the cabinets until all the caked-on dirt is removed.
Run a clean cloth under running water. Wring the excess water out, and use the cloth to wipe the surface of the cabinets, removing any cleaning powder residue. Dry the surface of the cabinets with a dry, clean cloth.