This phase I/II studies the side effects and best dose of natural killer cells before and after donor stem cell transplant and to see how well they work in treating patients with acute myeloid leukemia, myelodysplastic syndrome, or chronic myelogenous leukemia. Giving chemotherapy with or without total body irradiation before a donor peripheral blood stem cell or bone marrow transplant 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 and natural killer 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.

PRIMARY OBJECTIVE: I. To evaluate safety, tolerability and identify the maximum tolerated dose (MTD) of expanded natural killer (NK) cells to be used in patients with myeloid malignancies undergoing a haploidentical stem-cell transplant. SECONDARY OBJECTIVES: I. To determine survival of NK cells in vivo post-transplant. II. To determine the function of NK cells post-transplant and compare with a retrospective control treated with no NK cells. III. To estimate the proportion of patients with engraftment/graft failure. IV. To estimate the non-relapse mortality (NRM) at day 100 post-transplant. V. To estimate the cumulative incidence of grade III-IV aGVHD (acute graft-versus-host disease) at day 100. VI. To assess the rate of chronic graft-versus-host disease (GVHD) within the first year post transplantation. VII. To assess immune reconstitution post-transplant. VIII. To assess disease response, disease-free survival (DFS) and overall survival (OS) after transplantation. IX. To perform a retrospective comparison of patients treated on the study with NK cells will be performed with a Center for International Blood and Marrow Transplant Research (CIBMTR) control of similar patients who did not receive NK cells. OUTLINE: This is a phase I, dose-escalation study of NK cells followed by a phase II study. Patients are assigned to 1 of 2 conditioning regimens. MYELOABLATIVE CONDITIONING REGIMEN: Patients receive high-dose melphalan intravenously (IV) over 30 minutes on day -7, fludarabine IV over 1 hour on days -7 to -4, undergo total-body irradiation (TBI) on day -3, and NK cells IV over 30 minutes on day -2 or -1. NON-MYELOABLATIVE CONDITIONING REGIMEN: Patients receive melphalan IV over 30 minutes on day -7, fludarabine IV over 1 hour on days -7 to -4, undergo TBI on day -3, and receive NK cells IV over 30 minutes on day -2 or -1. TRANSPLANT: Patients undergo allogeneic peripheral blood stem cell (PBSC) or bone marrow transplant on day 0. POST-TRANSPLANT CYCLOPHOSPHAMIDE AND GVHD PROPHYLAXIS: Patients receive cyclophosphamide IV over 3 hours on days 3 and 4, tacrolimus IV beginning on day 5 for 2 weeks and then orally (PO) for approximately 4 months, and mycophenolate mofetil PO thrice daily (TID) beginning on day 5 for approximately 6-7 months. NK CELLS: Patients receive NK cells IV over 30 minutes on days 7 and 28-90. After completion of study treatment, patients are followed up for 2 years.