Remove the cabinet hardware with a small screwdriver. Put the screws in a safe place to prevent losing them.
Fill a small washtub with hot water and dish soap. To go a completely natural route, use a citrus cleaner or vinegar solution. Vinegar works best as a cleaner with a 50 percent water to vinegar ratio. For the citrus cleaner, follow the manufacturer's recommendations.
Soak the cabinet handles in the water for at least 30 minutes. This will give the cleaner time to cut the grease. Keep a kettle of water warming on the stove to add to the washtub once the water starts to cool. Very hot water works best.
Use rubber gloves to protect your hands and apply a soft sponge to the hardware to remove grease. Scrub particularly stubborn spots with an old toothbrush. Do not use harsh scouring pads, as these can scratch or remove the finish from your handles.
Rinse the hardware well and dry with a clean cloth or paper towel. Put the handles on a drying rack or lay on a dry towel to continue drying. You don't want any moisture to remain when you screw the hardware back into place. This could cause rust or mold to form underneath the metal.