Donor Stem Cell Transplant in Treating Patients With Previously Treated Lymphoma, Multiple Myeloma, or Chronic Lymphocytic Leukemia

OBJECTIVES: * Determine if allogeneic stem cell transplantation using unrelated matched or related haploidentical donor bone marrow or unrelated matched cord blood results in timely, complete, and durable engraftment in patients with previously treated lymphoma, multiple myeloma, or chronic lymphocytic leukemia. * Determine the incidence and grade of acute and chronic graft-versus-host disease in patients treated with this regimen. * Determine if the augmented graft-versus-tumor effect accompanying unrelated or partially matched donor allogeneic transplant reduces the incidence of relapse in these patients. OUTLINE: * Preparative regimen: Patients receive cyclophosphamide IV over 2 hours on days -7 and -6 and undergo total-body irradiation (TBI) twice daily on days -4 to -1. Patients who are unable to undergo TBI receive busulfan IV or orally 4 times daily on days -9 to -6 and cyclophosphamide IV over 2 hours on days -5 to -2. * Stem cell transplantation: All patients undergo unrelated matched bone marrow or umbilical cord blood transplantation or partially matched related allogeneic bone marrow transplantation on day 0. * Graft-versus-host disease (GVHD) prophylaxis: Patients receive GVHD prophylaxis comprising methotrexate and cyclosporine. Patients may be enrolled in other protocols directed towards GVHD prophylaxis.