If you're on a PC with Windows, download Windows Easy Transfer ... it's a free download from Microsoft for XP and I believe it's bundled in with Vista and Windows 7. Easy Transfer can back up your files to move to a new computer, and I believe this will include your saved game progress.
Which operating system and browser, and which versions of operating system and browser are you using? Do you have Javascript enabled in your browser?
Cormyn, you brought up an idea I had about my XP. I was thinking if I got a new computer, that I would want to keep all my games and files, so I would transfer them to the new computer. My question is, does that program transfer games that aren't flash, including like: Diablo, Starcraft, World Of Warcraft, well those types of games.
Matrix159, the Windows Easy Transfer software *can* back up program files, but generally it won't unless you specifically tell it to. It also likely won't be able to back up any settings in your registry, so you're really better off just reinstalling those games from scratch and downloading any patches. (which sucks for WoW, I know)