Ubuntu does not mount your real hard disk but you should be able to copy your data to an external hard easily enough.
First Boot from the USB and select try Ubuntu
I'm using the 12.04 CD to do this
Now press Ctrl+Alt+T to open a termininal
In the terminal enter the following command
sudo -i
Then
nautilus
This opens nautilus (the file manager) as root. Do not close the terminal.
Note: In the section down the left hand side of nautilus "File System" is the virtual file system of the live session not a real file system you real hard drives are shown as devices. In my case the device was "959cf23..."
As you are root you can now copy and paste any data you want to keep to external storage.
You can change ownership and permissions of the files by selecting properties
Alternatively don't worry about permissions for now as you can always change the permissions again from the other OS or you can make them all accessible to every one with
sudo chmod -R 777 /path/to/external/drive
Replacing /path/to/external/drive
with the correct path. Warning: Make sure you type this correcly as sudo chmod -R 777 / path/to/external/drive
for example will change permissions for the entire file system.
Then you can re-install the OS you want and restore your data from the backup.