Remove any handles or other hardware from the furniture with a screwdriver.
Fill a bucket with warm water. Add trisodium phosphate (TSP) cleaner according to the package directions. Use a sponge to wash the furniture piece with the TSP solution.
Rinse the sponge in clean water and use it to rinse the piece. Dry the furniture with a lint-free towel.
Sand the surface of the furniture with 220-grit sandpaper. Follow the grain of the wood. Stop sanding when the entire surface is rough to the touch, and the polyurethane sheen is gone. Wipe up the dust with a tack cloth.
Apply a coat of oil-based primer to the piece, following the grain of the wood. Use a foam paint brush to apply the paint in even, thin, overlapping strokes. Let the primer dry overnight.
Apply one coat of an interior oil-based paint in the same way you applied the primer. Let the first coat dry, and apply a second coat if needed. Let the furniture dry overnight.
Reattach the hardware.