Adobe does not yet have a 64 bit version of the AIR SDK or runtime for Linux. They do have some instructions for getting it to work. If you run Ubuntu, it is much simpler. just type:
$ sudo aptitude install ia32-libs lib32asound2 lib32gcc1 lib32ncurses5 lib32stdc++6 lib32z1 libc6 libc6-i386
I have only tested this on Jaunty, so I can’t promise it works on earlier releases, but it’s lots easier than the manual unpacking stuff on Adobe’s page.