I've been using that stiff plastic stuff - not adhesive, can't remember what it's called. Never had any problems with it provided it's the correct width and properly installed.
Beware of rim tape that's too narrow: I've had problems with two complete new bikes straight out of the shop (both, Bianchi road bikes, as it happens), on which, on cheapish double-walled rims with large, un-ferruled spoke holes, the tape was too narrow to cover the width of the holes! Result: mayhem. I believe Bianchi have sorted that particular problem, now.