ARM I: (experimental) Patients receive tacrolimus IV continuously or orally (when able) daily beginning on day -2 followed by a taper beginning on day 42 and continuing until day 98 (for patients undergoing matched sibling donor transplantation) OR tacrolimus IV continuously or orally daily beginning on day -2 followed by a taper beginning on day 100 and continuing until day 180 (for patients undergoing related, unrelated, or cord blood donor transplantation) in the absence of GVHD. Patients also receive methotrexate IV on days 1, 3, and 6 (for patients with matched sibling and umbilical cord blood donors) OR days 1, 3, 6, and 11 (for patients with unrelated bone marrow and peripheral blood stem cell donors) and oral sirolimus daily beginning on day 0 followed by a taper beginning on day 180 and continuing until day 207.