 This randomized phase III trial studies lenalidomide to see how well it works compared to a placebo in treating patients with multiple myeloma who are undergoing autologous stem cell transplant. Giving chemotherapy before a peripheral blood stem cell transplant helps kill any cancer cells that are in the body and helps make room in the patients bone marrow for new blood-forming cells stem cells to grow. After treatment, stem cells are collected from the patients blood and stored. More chemotherapy is then given to prepare the bone marrow for the stem cell transplant. The stem cells are then returned to the patient to replace the blood-forming cells that were destroyed by the chemotherapy. Biological therapies, such as lenalidomide, may stimulate or suppress the immune system in different ways and stop cancer cells from growing. Giving lenalidomide after autologous stem cell transplant may be an effective treatment for multiple myeloma. 