What is the best way to remove grease buildup on cherry cabinets?

Put vinegar on a soft cloth and wipe down the cabinets. Rinse with warm water, wring, and wipe again.

Mr. Klean gentle eraser.

A small amount of Dawn dishwashing soap added to the vinegar will help cut the grease as well

I use Old English to clean my cabinets twice a year. It is great and treats the wood. If there is A LOT of build up, the best cleaner is a degreaser from the Auto supply store. It will not hurt. I will give you a photo of it. Put it on a small area like 4 inches by 6 inches, let it set ten minutes, and scrub it with an old toothbrush. Wipe it clean with a damp cloth. Repeat as needed. It will get the worst grease off. Here is the de-greaser: