Remove the wood drawers from the cabinet or dresser. Place items from the drawers onto a bed or table. Attach the brush attachment to your sweeper.
Vacuum inside and outside the wood drawers to remove surface dust. Flip the drawer upside down and vacuum the underside as well.
Moisten a clean cloth with 2 tsp. mineral spirits. Wipe the inside and front of the wood drawers with the cloth. Scrub into crevices or around handles with an old toothbrush dampened with the mineral spirits to remove dirt from hard-to-reach places.
Wipe excess mineral spirits from the wood with a lint-free cloth. Repeat the procedure on each wood drawer. Replace items back into the drawers and place the drawers back into the dresser or cabinet.