This phase I/II trial studies whether stopping cyclosporine before mycophenolate mofetil is better at reducing the risk of life-threatening graft-versus-host disease (GVHD) than the previous approach where mycophenolate mofetil was stopped before cyclosporine. The other reason this study is being done because at the present time there are no curative therapies known outside of stem cell transplantation for these types of cancer. Because of age or underlying health status, patients may have a higher likelihood of experiencing harm from a conventional blood stem cell transplant. This study tests whether this new blood stem cell transplant method can be made safer by changing the order and length of time that immune suppressing drugs are given after transplant.