Having a T-Mobile Samsung Galaxy S3 rooted is a 5-minute procedure (including SGH-t999) that requires micro USB wire and a computer along with your phone.
Here is a few simple steps demonstration
Hold and long press Volume Down, Home Centre and Power buttons. After a few seconds, an option will appear to get into download mode.
Now press Volume Up button and this will initiate download mode.
Connect to your computer after download mode has started.
Next, you need to download GalaxyS3RootSprint.zip
Unzip it and extract files.
Open odin3 v1.85.exe to start the app. A yellow highlighted box with a COM number will appear on computer screen. If it doesn’t, your drivers are not installed correctly. In other case Kies also interfere sometimes. So make sure to close it in Task Manager before you proceed.
Open GalaxyS3RootSprint folder and select PDA. Then choose clockworkmod.tar file and click Start.
** Wait for a few seconds until reboot and shows ClockworkMod Recovery completion.
** PASS dialogue will appear if the procedure went correctly.
After reboot, open the folder GalaxyS3RootSprint and copy the file CWM-SuperSU-v0.87.zip to the internal storage or SD card of your phone.
Now just enter ClockworkMod Recovery by long pressing Volume Up, Center Home, and Power button together for almost 10 seconds.
Now select “install zip from SD card”.
After this select “choose zip from SD card” or “choose zip from internal SD card” depending on where you have copied the file in step 8.
Now select CWM-SuperSU-v0.87.zip and click on Yes.
When the Superuser zip file CWM-SuperSU-v0.87.zip stops flashing, reboot and here you go with fully rooting phone without any changes in your settings.
** SuperSU app will appear once you have rebooted and can enjoy unlimited apps now.
Make sure to check out my Galaxy S3 page for future updates.