Living Organ Donor Job Security Act - Amends the Family and Medical Leave Act of 1993 (FMLA) to entitle employees covered by FMLA to leave to provide a living organ donation, including for time spent for: (1) tests to determine medical suitability of the employee for donation; (2) physical, psychological, and social evaluations of the donor; (3) pretransplant outpatient services; (4) postoperative inpatient and outpatient transplantation services; (5) travel in connection with such tests, evaluations, and services; and (6) recuperation.