Donor Umbilical Cord Blood Transplant With or Without Ex-vivo Expanded Cord Blood Progenitor Cells in Treating Patients With Acute Myeloid Leukemia, Acute Lymphoblastic Leukemia, Chronic Myelogenous Leukemia, or Myelodysplastic Syndromes

Inclusion Criteria: * Age criteria: * High dose TBI regimen: 6 months to =\< 45 years * Middle intensity TBI regimen: 6 months to =\< 65 years * Conditioning regimen selection should be based on the underlying disease, presence of minimal residual disease (MRD), age, co-morbidities, attending physician, and site preference; conditioning regimen will not require stratification of the randomization due to heterogeneity in the cohort of eligible patients. * Acute myeloid leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia * All patients must have acute myeloid leukemia (AML) that is considered best treated by stem cell transplant by the referring physician and the attending transplant physician * All patients must be in complete remission (CR) as defined by \< 5% blasts by morphology/flow cytometry in a representative bone marrow sample with cellularity \>= 15% for age * Patients in which adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the principal investigator prior to enrollment * Acute lymphoblastic leukemia, including biphenot…