Kichler has a new line of very thin LEDs. I'd guess they are about 1/2" thick. But they get pretty hot. I normally think of LED as cool, I think it must be the transformer/driver. There are "linear lighting" products that are essentially landscape lighting technology that mounts under your cabinet. You can attach light fixtures along the way almost where ever you wish. Xenon, LED, Halogen. Seagull makes one version. The light sockets are 3/4" thick. Personally I don't see anything wrong with a light valance. The valance doesn't have to be very high. I plan to add one to my cabinets to help hide the lights. 3/4" isn't very much. I think mine are about the same. Here is a link that might be useful: Seagull Linear Lighting

Besides fishing wire, there are a number of other things to consider such as 1. Location of the switch(es) - new switch boxes? 2. Physical structure of the cabinet bottoms - frameless european style without intervening styles or not. 3. Type of lights - Kichler vs eW Profile vs ? T4/ T5 fluorescent I've included a link with a number of details Here is a link that might be useful: UCL continuation

You'd need to know if the boxes when joined together form a contiguous flat bottom for each cabinet run. Figure out if you're going to use 120V AC or low voltage lighting. The following thread has more information. Here is a link that might be useful: DIY LED UCL continuation

It's kind of personal preference. Most of the new showcase homes by me are all using 12v LED ribbon since it's inexpensive, easily concealable, and produces even light. Right now I have 110v fixtures (really it transforms it to 12v) fixtures with LED pucks. Mine are somewhat even but I can tell it's a focused source... My big fixtures are easily changeable, but I think I'll just go with the thin ribbon next time