Question: So, I got a fancy new computer and I'd like to move adobe flash's saved data from computer a to computer b.

Side note: computer a is win 7 64 bit, and computer b is ubuntu 64 bit. However, I may fall back to win 7 so answering with the generic process of backing up and restoring the saved data is great.

Answer: 
You are asking how to transfer flash cookies from one PC to another. That's answered in a previous SuperUser question: [Folder location of Flash cookies on Windows Vista](https://superuser.com/questions/31457/what-folder-contains-flash-cookies-on-vista). Just copy the contents of the appropriate folder from one PC to the other (while your web browser and any Adobe AIR applications are NOT running) and all should be well.

If the data is stored on a server, instead of your PC, obviously this won't help, but that folder is the only place Flash Player is permitted to store its persistent information.
If you're asking for what I think you're asking for, cached Flash files downloaded from the web, you'll want to look through your cache directory for `*.swf` files, and move those to wherever Machine B expects to find them.

I could, however, be misunderstanding the question ...