I removed all of the doors and painted them separately from the cabinets. It took a full four coats of paint for a completely uniform appearance that covered up the old stain color (which I like to call "Orange Oak.") Confession: I skimped and only did three coats on the inside of the doors... you rarely see them anyway!
Without further ado, here are some more progress photos of our newly-white cabinets:
After four coats (!) of chalk paint