Sweep the top of the cabinets with a whisk broom. Collect all of the loose dust into a dust pan and discard it.
Fill a bucket with warm water. Add a non-toxic household cleaner rated for your cabinets' materials. For example, use an oil soap for wood cabinetry. Pour in the cleaner according to the package directions to mix it with the water.
Dip a scrub brush into the solution. Scrub the top of the cabinets, starting in one corner. Scrub a 2-foot section until the sticky residue is gone.
Wipe up the moisture with a clean, dry cloth. Scrub the next 2-foot section and wipe it dry once the sticky residue is removed.
Empty the bucket and rinse it out when the water begins to look dirty. Refill it with clean, warm water and cleaner. Continue scrubbing the cabinets in 2-foot sections then drying, as you go, until you've cleaned all of the cabinets.