When we decided to use beadboard panelling, we first thought we'd just do the backsplash area. But when we realized we could benefit from a lot more color in the kitchen, we decided to extend the panelling all the way around the room.
After cutting the panels down to size and painting them, we began installing them on the wall using construction adhesive and finishing nails.
Construction adhesive on the wall:
Panelling going up on the wall that will be behind the refrigerator (hence the bits-and-pieces approach -- it won't be too visible). Note our inept cutouts for the electrical outlets (we got better at this part).
Backsplash in place:
Finished panelling, with seems caulked and awaiting paint touch-ups and trim:
Cheerful, sunny, and bright!