This randomized phase III trial is studying total-body irradiation (TBI) and fludarabine phosphate to see how it works compared with TBI alone followed by donor stem cell transplant in treating patients with hematologic cancer. Giving low doses of chemotherapy, such as fludarabine phosphate, and radiation therapy before a donor stem cell transplant helps stop the growth of cancer cells. It also stops the patient's immune system from rejecting the donor's stem cells. The donated stem cells may replace the patient's immune system cells and help destroy any remaining cancer cells (graft-versus-tumor effect). Sometimes the transplanted cells from a donor can also make an immune response against the body's normal cells. Giving cyclosporine and mycophenolate mofetil after transplant may stop this from happening. It is not yet known whether TBI followed by donor stem cell transplant is more effective with or without fludarabine phosphate in treating hematologic cancer.