What does a jet engine technician do?
You ask a very broad question. Jet engine technicians can work in various capacities depending on the job they are hired to do.

In my experience, I worked on the flight-line directly on aircraft removing/installing, performing operational checks, troubleshooting and servicing engines and other systems.  

Some technicians work on engines out of the aircraft and in a controlled environment like the factories where engines are made. Others work in repair shops tearing down and rebuilding existing engines.