This phase II trial studies how well donor peripheral blood stem cell (PBSC) transplant works in treating patients with hematologic malignancies. Cyclophosphamide when added to tacrolimus and mycophenolate mofetil is safe and effective in preventing severe graft-versus-host disease (GVHD) in most patients with hematologic malignancies undergoing transplantation of bone marrow from half-matched (haploidentical) donors. This approach has extended the transplant option to patients who do not have matched related or unrelated donors, especially for patients from ethnic minority groups. The graft contains cells of the donor's immune system which potentially can recognize and destroy the patient's cancer cells (graft-versus-tumor effect). Rejection of the donor's cells by the patient's own immune system is prevented by giving low doses of chemotherapy (fludarabine phosphate and cyclophosphamide) and total-body irradiation before transplant. Patients can experience low blood cell counts after transplant. Using stem cells and immune cells collected from the donor's circulating blood may result in quicker recovery of blood counts and may be more effective in treating the patient's disease than using bone marrow.