The first thing I would try is look in your kitchen cabinets at the ceiling on both sides and see if you have a wire that looks like it is just screwed into the paneling or ceiling trim. Loosen and then tighten that screw and see if your lights work, if not then more troubleshooting needs to be done with a length of wire. Take the lens from one light that works and one that doesn't work, turn all the lights on that don't work, touch one end of the wire to the ground on the working light (the outside of the bulb is the ground) and the other end to the non working light and see if the lights come on. If they do you have a grounding problem, if not you probably have a power problem. If I was a betting man I would bet you have a grounding problem above your kitchen cabinets. Check wire nuts above cabinets. Don't worry about getting shocked, you're dealing with 12 volts not 120.
PS I had the same issue and it was the grounding screw on the door side above the cabinets.