So you've decided to run a 64-bit edition of Linux on your computer but noticed that Flash is still in its 32-bit phase because in order for it to be 64-bit compatible would require a whole rewrite of Flash, and therefore isn't running on 64-bit Firefox. Don't worry, you can still run 32-bit Flash on 64-bit Firefox, you just need to install a couple of things. The two programs you're looking for is nspluginwrapper and cdbs. After installing these programs, you may have to restart Firefox for it to take effect.

If it started working for you but later stopped, it could be because it crashed at some point (in the background) without informing you. To get it working again, restarting Firefox may help.