How to run low voltage wire from one side of kitchen hood to the other?
I've done this myself with both conventional range hoods and over-the-oven microwaves. Just drop the hood and cut one in. Avoid the areas where mounting screws will need to go, and avoid damaging your vapor barrier (if present). There's no need to patch the drywall. 

Use any rigid or semi-rigid tubing that meets your size requirements. Obviously you'll want to position it so that it aligns with the holes you drill in your cabinetry.

UPDATE: 
From your comment I see that you'd like the wire to be under the upper cabinets. This is what I expected. My wire route usually goes through the side of the cabinet adjacent to the range hood, down through the cabinet floor at the rear corner, and wherever it needs to go underneath the cabinet, concealed by the recess. Sometimes it's necessary to clip a very small corner off the floating shelves, but it's not really visible from outside.

Of course, you can always run your conduit in the wall level with the cabinet recess since you're working on the wall anyway. The "right way" is what makes sense in your case. It's not a code issue.