Yes, the wires can be run through rectangular "plastic cable raceway" (see below). If you want a finished look, buy some "end caps" to put on where the cable raceway ends rather than leaving a razor cut across. Couldn't quite get my electrician or GC to understand the need for that, so did it myself.

Here is a link that might be useful: Cable Raceway for UCL's

SparklingWater,

Thank you for your suggestion.

I ran all my wires underneath the cabinets then hid them with wood cut out to match the underside of each cabinet, then glued small rectangular pieces under the cabinet edges and screwed the wood with tiny screws into the rectangular pieces.Then I stained the wood to match the faux wood grain under the cabinets, then cut out circles to mount the pucks into so all wires would be hidden for a nice clean custom look.

So, you don't have to run wires inside the cabinets.