¿Es posible rootear el Xperia t3?

Sí es posible, he encontrado este método entre muchos otros.

Advertencias:

A continuación se presentan algunas advertencias que debe tener en cuenta, antes de proceder con el tutorial del rooting:

Rooting your Sony Xperia T3 Android Smartphone will void its warranty. 
Hence, to get back warranty after rooting your Sony Xperia T3, you will 
have to flash the stock ROM back again.

Take a complete backup of all the data stored on your Sony Xperia T3, 
as the rooting process will wipe away all the data stored on your Smartphone.
There are risks of your Sony Xperia T3 getting damaged, if the rooting guide below 
is not followed properly. Hence, proceed at your own risk only and we are
not to be blamed in any such case scenarios.

Requisitos previos:

Aquí hay algunos procedimientos de requisitos previos para completar antes de rootear Sony Xperia T3 Android Smartphone:

Download and install the latest USB Drivers for Sony Xperia T3 on your PC.
Charge your Sony Xperia T3 until it has a minimum battery backup of 80%.
Download the root tool from here.
Enable USB Debugging on your Smartphone.

Root Sony Xperia T3 fácilmente usando el método de un clic:

A continuación se muestra la guía detallada sobre cómo Root Sony Xperia T3 Android Smartphone fácilmente utilizando el método de un solo clic Root:

Connect your Sony Xperia T3 to the PC.
Copy/Paste the root tool that you downloaded earlier to your Smartphone.
Disconnect your Smartphone from the PC.
Now, navigate to the copied file using file explorer, and click on the copied file once.
This will begin the process to install the root tool on your Sony Xperia T3.
After installation, launch the app from your Smartphone.
Follow the on screen instructions and when prompted to choose 
an EXPLOIT, choose the “GANDALF” exploit.
Wait for the rooting process to complete.
Reboot your Sony Xperia T3.

Espero que esto ayude.

Fuente : Rootandroid