Can I integrate my over-the-range microwave hood's lights with my under cabinet lights?
Electrically speaking there is no reason why you can't hook up those lights. But the reason I wouldn't has to do with the warranty of the microwave, and let's suppose you try and connect them up and something gets lost or broken or even worse now the microwave doesn't work. No manufacturer is going to take that microwave back and service it once it has been tampered with. You might be better off just installing a strip of lights attached to your existing under counter lights and be done with it.