An electrical submersible pump motor has motor oil flowing through external circulation tubes for cooling the motor. A substantial portion of the exterior of each tube is submerged in and exposed to wellbore fluid. Heat is transferred from the motor to the motor oil, and then circulated through the external circulation tubes to conduct heat to the wellbore fluid. Internal or external motor oil pumps may be used to propel the motor oil through the circulation tubes. Guards or baffles may be used to protect the circulation tubes and to influence the flow of production fluid over the circulation tubes.