Fill your sink with hot, soapy water. Use liquid dish soap or any other household detergent that isn't abrasive. Scrub the surface of the cabinets and cabinet doors with a nylon scrub brush. Wipe them with a damp cloth to rinse, and let the cabinets air dry.
Remove the cabinet doors from the cabinet housing. Remove all the hinges and handles. Lay the cabinet doors face up on a plastic tarp. Cover the counters and backsplash with drop cloths.
Fit a hand-held electric sander with a 220-grit silicon carbide sandpaper. Sand the cabinet housing and the cabinet doors, following the grain of the wood, until the shine from the polyurethane finish is gone. Wipe the cabinets and doors with a tack cloth to remove the dust.
Apply an even coat of brushing putty to the cabinet doors and housing using a nylon-polyester paintbrush. Use the brushing putty instead of oil-based primer on oak cabinets because it fills in the open grain.
Let the first coat dry according to the package directions. Use the electric sander to sand down the first coat until it is smooth. Throw away the paintbrush and use a second, clean one to apply the second coat of putty. Let it dry, and sand it smooth.
Apply the first coat of semi-gloss latex paint to the cabinets and doors with a polyester paint roller. Use even, overlapping strokes. Use a paintbrush to apply paint to the inside corners of any trim that the roller can't reach. Let the first coat dry according to the package directions, then apply one more coat.
Wait 3 days. Flip the cabinet doors over and apply two coats to the back in the same way you applied the paint to the front of the doors. Wait 3 days.
Re-attach the hardware to the doors. Re-attach the doors to the cabinets. Remove the drop cloths.