Assuming that the flavor of Unix is staying the same (AIX?) then the operation is fairly simple. Naturally, the Teamcenter envirnment must be taken down before making any changes to the underlaying architecture.
A rough outline:
Take down the Teamcenter environment and lock it.
Export the database from the original DB server.
Import the database to the new DB server.
Test the DB connection and regen the POM_SCHEMA.
Unlock Teamcenter and start it back up.
Modifying TC_DATA is the "secret sauce" that tells Teamcenter how to connect to the database.