I used 110's on my last job with these.
I usually run the cords on the backside of the lip underneath the cabinets, and drill holes to pass the cords through between edges. Bring all the cords up into the cabinet with the controller/xformer.
But unless your cords are listed for in-wall, that, or possibly (if the cabinets are mounted on plates/boards on the wall) in the void behind them, but that's more for new work.
