Now confirmed on WiFi model AND Verizon Model (thanks Rockaholic94 and badogg for testing).
1. Download this file and rename it as "update.zip"
2. Put the update.zip file on the root of your tab's SD card
3. While your tab is powered on or off (doesn't matter), hold the volume down (volume up on VZW Model) + power key to bring up the option to select recovery mode or download mode.
4. Choose the recovery mode option (box picture) by hitting the volume down key then selecting with the volume up key.
5. Now that you're in the factory android recovery menu, choose "Install update from SD card".
6. Highlight the update.zip file using the volume buttons and the power button to select.
7. Confirm Superuser is in your list of apps
Let it do its thing and it is now rooted.
1. After confirming the Superuser app is installed and you in fact have root access, download ROM Manager from the Play Store.
2. Open ROM Manager and choose "Install Clockworkmod Recovery"
3. Select "Samsung Galaxy Tab 10.1" to make sure the right recovery is installed, and wait a minute or so as it installs.
4. Towards the end, ROM Manager will prompt you for root access to complete the install.
Congrats, you now have root access and CWM Recovery.
This was tested on my 3G device running 3.2 for the record, but it should work on 3.1 also. Please let me know what devices you've been successful using this method for, and remember the "thanks" button too .
Guide was written by myself, but I grabbed the root file from this post on Android Authority. Props to whoever the original owner of that file is... Also, not responsible for bricking devices etc etc etc.