How do I reapply travertine tile used for a backsplash on my kitchen wall
With travertine you should be using thinset to apply it to the wall.  You don't really need to grout - I have seen it done either way travertine.  If you don't grout then make sure you use a darker grey thinset as it cures into more of a mortar look and also make sure that the thinset isn't squirting up out of your tiles.  So there is a little cleanup to do after applying the thinset.