Help!

I have cabinets with no face frame, and so I am looking for an under-cabinet lighting solution that minimizes the light's profile. from left to right there is a cabinet, vent hood with short cabinet over it, cabinet, and last is the fridge. the cabinets on either side of the stove need the lights.

I would prefer not to go into the wall to hard wire anything. I puchased a set of 120v puck style lights at lowe's that can be recessed. they have one main lamp cord that i thought i could plug into the open outlet behind the fridge. the xenon lights have their own leads that can be attached anywhere along the lamp cord's length. I thought I could recess the lights, run the lamp cord through the cabinets and cover the cord by creating a false bottom in the cabinets. HOWEVER, I read the fine print that said lamp cord can be run in the cabinets but should not be run from cabinet to cabinet via small drill holes!

1) can i run lamp cord through kitchen cabinets??

2) if not, how can i create attractive low profile lighting without a tangle of wires under my cabinets without adding trim?

Any input is greatly appreciated, thanks!!