I didn't expect this to be so hard, really. The rooting is usually a pretty straight forward process that can be done quickly. However, considering the amount of issues around the upgrading ICS table to JB with or without rooting the first; number of the posts on xda-developers.com forum that refers to the same partially outdated recovery images, it wasn't easy. However, here's the easiest way to root your stock JB Transformer device:
# I am doing everything on Linux
# You need to either install Android SDK or get adb and fastboot tools for your distribution
1. Push CWM-SuperSU to the device
% adb push SuperSU.zip /sdcard
2. Download and unpack ClockWorkMod Recovery v220.127.116.11
rename recovery_jb.img to recovery.img
3. Boot your device to fastboot mode
% adb reboot bootloader
Use Vol- to scroll to USB icon (fastboot mode); use Vol+ to select
4. Unlock the bootloaded using UnLock_Device_App_V7.apk (google to download the file). Or alternatively you should be able to do use
% fastboot <VendorID> oem unlock
(please note, that you might be better off by running fastbook as root user). ASUS VendorID is 0x0B05. To find out the id for your device use lsusb.
5. Flash recovery.img to your device and reboot
% fastboot -i 0x0B05 flash recovery recovery.img
% fastboot -i 0x0B05 reboot
6. Boot to fastboot mode again (as in #2 above) and enter Recovery mode
7. Install SuperSU from the zip file on sdcard
8. Reboot once again
9. Install RootChecker from Google Play and make sure your devices is rooted.
"The bank killed me"
11 months ago