Things like Blind Nuts work great for attaching bumpers.
http://www3.towerhobbies.com/cgi-bin/wti0001p?&I=LXK019
The little tangs you see sticking up get pressed (re: Hammered) into the wood, and hold very well.
These require access to whatever you are mounting the bumper to from the other side, so because of this aspect, it may not work for everyone.