This is a step-by-step tutorial with screenshots.

On The PlayBook

Plug your PlayBook into your Linux computer using a USB cable. You may see this screen (or similar) you can dismiss it.

In the Settings menu, scroll down to "Storage & Sharing".

Ensure that the "USB Connections" is set to "Connect to Windows".

Scroll down and make sure that "File Sharing" is set to "ON".

Finally, still in settings, scroll up the left panel to "About" and select "Network" Make a note of the USB IPv4 Address (in this image it is 169.254.144.217).

On Ubuntu

Open a terminal and type

sudo ifconfig -a

You may be prompted to enter your password.

Scroll through the results and you should see an entry of "usb0".

To connect to the PlayBook, click on "Places" then "Connect to server".

Select the server type to be "Windows Share". Enter the IP address of the PlayBook (found from the PlayBook's About menu).

If all has gone well, you should see this screen:

Click on "media" and you'll have access to all of your PlayBook's storage space.

That's It!

You should now be able to copy files back and forth. Unlike a normal USB disk, you don't need to eject or unmount the drive once you've finished copying files to the device. Note - when you disconnect the USB cable and reconnect it, the IP address may change. You will need to go to the PlayBook's network settings to find the new IP address. I can't find a way to get a static IP. If you know how - or have any other questions - please leave a comment.

Update

You can set a static IP by turning on Development mode, or follow these instructions from Jarviser on the BlackBerry Support Forums

Forget the IP address, use Tablet Network Name which will always be the same...

In Storage and Sharing, after selecting Connect to Windows, touch Properties button.

You will find

Tablet Network Name - Use that instead of IP Address in Ubuntu. (Mine says PLAYBOOK-24B6) You will then be asked for Username and Password, Mine is "playbook" and the playbook's sharing password. WORKGROUP is left unchanged.