How to attach a wood moulding trim around a backsplash
Construction adhesive would certainly do the job, but it would destroy the drywall when you remove it. 

I'd use silicone caulk as both the adhesive and the caulk to the countertop. It'll be rock solid, and if you ever need to remove it you may be able to do so without ripping paper off the drywall. Just put 2 or 3 quarter-sized dabs along the back. They'll flatten out to create nice wide attachment pads. 

Be sure to completely paint or seal the trim, front and back, before attaching. Water is a tough nemesis for wood.