This randomized phase III trial studies how well graft-vs-host disease (GVHD) prophylaxis works in treating patients with hematologic malignancies undergoing unrelated donor peripheral blood stem cell transplant. Giving chemotherapy and total-body irradiation before a donor peripheral blood stem cell transplant (PBSCT) helps stop the growth of cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from a donor 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. Giving total-body irradiation (TBI) together with fludarabine phosphate (FLU), cyclosporine (CSP), mycophenolate mofetil (MMF), or sirolimus before transplant may stop this from happening.