Do your cabinets have molding (example) that is attached to the bottom of the cabinet box to hide any lights and wires you might attach there? If not, I'd use tape lights in a channel as that will be the shallowest solution (hide the required driver inside a cabinet). Otherwise, linkable light bars shallower than the molding would be a good solution. Be sure that whatever solution you choose is dimmable from a wall dimmer. I'd skip the lights under the cabinets. They'll just highlight the dirt on the kitchen floor.

Ours are a fixture mounted into a groove routed into the molding below the cabinets like wdccruise suggested. Very sleek - you can’t see them at all from the front, and there’s no ugly light bar under there.

@Candace's solution is elegant but hiding the lights behind the molding is sufficient.

If you choose an under-cabinet light with its own dimmer or color temperature selector and you'll turn it on and off with a wall switch, be sure the light does not "forget" its brightness and/or color temperature settings when switched off and on from the wall switch..

LED strips be the best variant for kitchen!

We like using the tape light!!!! Light bars are clunky and outdated, IMO. Also if you have any glass doors, its so easy to illuminate the entire cabinet behind the frames.