This phase II trial is studying how well umbilical cord blood transplant from a donor works in treating patients with hematological cancer. Giving chemotherapy and total-body irradiation (TBI) before a donor umbilical cord blood transplant helps stop the growth of cancer and abnormal cells and helps stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from an unrelated donor, that do not exactly match the patient's blood, are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. Sometimes the transplanted cells from a donor can make an immune response against the body's normal cells (called graft-versus-host disease). Giving cyclosporine and mycophenolate mofetil before and after transplant may stop this from happening.