How to clean Tacky, sticky kitchen cabinets?

It is a build up of grit and grime over time. Use a paste of baking soda and water and a magic sponge and clean all the cabinets well. Remove the handles because it's hard to clean around them. Clean handles in hot soapy water. But put them all back on the original cabinet, if they're handles and not knobs. They may have been drilled differently to fit. Good luck.

KrudCutter for Kitchens -- works like magic on greasy build up -- any home improvement store will carry it

Sounds like just years of grease and dust accumulating. The very best cleaner I have ever used for that is good old ammonia. Pour some in some hot water, and just start scrubbing away... be sure to wear glove and have some good ventilation because that stuff stinks. But boy, does it work well. Cuts through that junk and dissolves it right away. If they're really dirty, you many need to go back with a second clean bucket of water and ammonia to clean leftover residue off. But you'll see how after just minute of the ammonia water soaked sponge hitting the greasy mess, it loosens it beautifully!

Ahhh.... Besides the magic eraser feeling like it's going to remove the paint, does the paint feel like it comes off easily? If not, and it seems like it's stuck to the previous layer, I think I'd just repaint them. I'd use THIS PRIMER and THIS PAINT. The primer I'm suggesting you use is a bonding primer and water clean up. It sticks way better than just standard primer. The paint I'm suggesting is a great quality satin finish paint. I'm just recently become a Behr paint 'fan'... painted my entire house inside and outside with it. I love how it flows out, hiding brush strokes. Hope that helps. Liz SimpleDecoratingTips