This phase II trial is studying how well rituximab works in preventing acute graft-versus-host disease (GVHD) in patients undergoing a donor stem cell transplant for hematologic cancer. Giving chemotherapy and total-body irradiation before a donor stem cell transplant helps stop the growth of cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. The donated stem cells may replace the patient's immune 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 a monoclonal antibody, rituximab, together with anti-thymocyte globulin, tacrolimus, and mycophenolate mofetil before and after the transplant may stop this from happening