"The easiest way for the S3 I9300 (international Version) and the S3 I9305 (international Version LTE) is to use krazyroot.
You can find the guide here:
KrazyRoot.com - Learn How To Root Galaxy S3 and Install Custom ROMs
To sum it up, you just have to install the apk and run it.
This installs SuperUser and the su binary. That's it, you are rooted!
Another way is using the chainfire root method.
[03.07.2012] CF-Root 6.4 - SuperSU+CWM5.5+CW… | Samsung Galaxy S III I9300, I9305
You'll need odin and the rootkernel(download button at the very bottom of the page).
Now, make sure your phone is charged. Kill all Kies processes in the taskmanager.
Copy pasta from the xda thread:
Unzip the attached Odin3-vX.X.zip file
Download and unzip the CF-Root-SGS3-vX.X.zip file (see posts below this one)
If you end up with a recovery.img and cache.img file, you've extracted twice. You need to end up with a .tar file - don't extract that one
(USB) Disconnect your phone from your computer
Start Odin3-vX.X.exe
Click the PDA button, and select CF-Root-SGS3-vX.X.tar
Put your phone in download mode (turn off phone, then hold volume-down + home + power to boot - if it asks you to press a button to continue, press the listed button, or run adb reboot download command)
(USB) Connect the phone to your computer
Make sure Repartition is NOT checked
Click the Start button
Wait for Android to boot
Done!"