Restore synchronized files/folders from Ubuntu One
All you need to do is sign into Ubuntu One in your new install of Ubuntu. This will start the syncing process to your new installation.
The files may take a little time to appear in the Ubuntu One folder, but they will eventually appear. There are ways you can check the syncing progress too from the indicator.
This also works for any other computer you might have too, say if you want to sync files between two machines.