They have little discs that adhesive mount to the dash that you then suction onto. Had one on the dash of my neon for a while, but when I eventually removed it, it took the colored layer of the dash pad with it....
There are also the sandbag mounts that have a similar disc in the middle, whole thing stays in place with their weight and non skid base.
Instead of a damp rag, spit into the suction cup first. But it will eventually fall off, no matter what you do. Esp after sitting in hot sun for a few days.