SECOND I AM NOT RESPONSIBLE IF THIS WILL BRICK YOUR PHONE!
Under normal circumstances it won't. If something goes wrong you can still flash original firmware with odin.
THIRD this can only be done in Windows.
Now the procedure
1. Download GT-I9070-root.rar, and extract it somewhere on you local disk.
2. From stuff you just extracted run program Odin3 v1.83.exe
3. Then open Settings in your phone, go to Aplications -> Development and check USB Debugging and Allow mock locations. Both needs to be Checked (Selected).
4. Turn off phone and turn it back on into Download mode, like this press Volume Down + Home button + Power button, wait untill some "strange" screen appers, put in usb cable and choose continue.
5. Now wait until Odin recognize your phone. It can take a while first time, if you have problems search for samsung drivers on the internet,
6. Under PIT file browse for GT-I9070_EUR_XX_8G_20111220_KL6.pit, you have extracted this file before.
7. Under PDA choos rootkernel.tar.md5.
8. Then check what options are checked (Selected), be sure to UNCHECK (remove selction) Re-Partition, Flash Lock, Phone EFS Clear, Phone Bootloader Update! Only Auto Reboot and F.Reset Time should be checked (Selected)
9. When this is set just press Start and wait for phone to reboot.
10. Now repeat steps from 4 to 9, with one exception at step 7 now choose xxld1.tar.md5.
11. Download and install Superuser or SuperSU aplication from market and enjoy the rooted phone! Or you can use a Doomlords rooting tool or Superoneclick, both this tools push the Superuser aplication to /system/app, which might be even better. Just search this tools on XDA.
Voila your phone should now be rooted. I will do my best to improve this method.
This wouldn't happened without a help of this people (THANK YOU GUYS!):
Your information were priceless!!