Donor Umbilical Cord Blood Stem Cell Transplant in Treating Patients With Hematologic Malignancies

Inclusion Criteria: * Patients will be diagnosed with one of the following hematological malignancies: acute myelogenous leukemia (AML), acute lymphoblastic leukemia, non-Hodgkin's lymphoma, myelodysplastic syndrome (MDS), chronic myelogenous leukemia (CML), and myeloproliferative and lymphoproliferative disorders * AML--First remission (CR1) with high risk features including a known prior diagnosis of myelodysplasia (MDS); therapy related AML; white cell count at presentation \> 100,000; presence of extramedullary leukemia at diagnosis; unfavorable AML subtype (M0, M5-M7); poor cytogenetic markers (abnormalities of chromosome 5, 7 or 8, 11q23, Philadelphia chromosome, complex karyotype) * AML--Second remission (CR2) or subsequent remission * AML--Relapse/Persistent Disease with \< 20% bone marrow blasts * ALL--First remission (CR1) at high risk for relapse as defined by: B cell ALL white blood cell count (WBC) at presentation \> 30,000 (T cell ALL WBC \> 100,000); presence of high-risk cytogenetic abnormality such as t(9;22), t(1;19), t(4;11) or other MLL rearrangements (11q23), t(8;14) * ALL--Second remission (CR2) or subsequent remission * ALL--Relapse/Persistent Disease with \< 20% bone marrow blasts * Non-Hodgkin Lymphoma--Induction failure or relapse and sensitive to most recent chemotherapy * MDS--Low or Intermediate-1 International Prognostic Scoring System (IPSS) score with: life-threatening cytopenia(s); and/or red cell or platelet transfusion dependence * MDS--ANC \< 500, recurrent infections, PRBC transfusions \> 2 units/month, poor risk cytogenetics, platelet transfusion dependence * MDS--Intermediate-2 or High IPSS score * CML--Chronic phase I (CP1) and resistant to or intolerant of tyrosine kinase inhibitors (i.e. imatinib, dasatinib, etc.) * CML--CP2 or subsequent chronic phase, including chronic phase achieved after induction therapy for blast crisis * Myeloproliferative and lymphoproliferative disorders--eligibility to be determined by a consensus of the physicians on the Case Comprehensive Cancer Center Leukemia/Lymphoma Multidisciplinary Committee * Myeloproliferative and lymphoproliferative disorders--must have evidence of disease acceleration to be a candidate for umbilical cord blood transplant; myeloproliferative disorders eligible for transplant include chronic myelomonocytic leukemia (CMML) with high IPSS score and myelofibrosis * Myeloproliferative and lymphoproliferative disorders--potential lymphoproliferative disorders eligible for transplant include chronic lymphocytic leukemia, prolymphocytic leukemia, and large granular lymphocytic leukemia * Good performance status: Karnofsky \>= 70 % or ECOG 0-1 * Calculated creatinine clearance \>= 60 mL/min, or measured creatinine clearance \>= 60 mL/min (by 24-hour urine collection) if creatinine \>= 1.5 or history of renal dysfunction * Hepatic Transaminases \< 4 x upper limit normal (ULN); total bilirubin \< 2.5 mg/dL, unless the patient has a history of benign congenital hyperbilirubinemia (Gilbert's syndrome) * Normal cardiac function by echocardiogram or radionuclide scan, (left ventricular ejection fraction \> 45%); if the left ventricular ejection fraction is between 40-50%, clearance by an adult cardiologist is required * Pulmonary function tests demonstrating FEV1 \> 60% of predicted for age * Adults must have a DLCOva \> 60% normal * For patients unable to complete pulmonary function tests clearance by an adult pulmonologist is required * Patients will be eligible for the clinical trial under the following conditions: they do NOT have an HLA-A/B/DR B1 identical RELATED bone marrow donor; they do NOT have a 6/6 HLA-identical matched unrelated adult donor; OR a matched related donor transplant is not in the best interest of the patient (i.e., patient's condition precludes waiting on the donor, too much time to prepare the donor, the donor is ineligible due to medical reasons, or in the case of high risk disease a related donor is not appropriated (syngeneic transplant); the decision must be agreed upon by the consensus of physicians on the Case Comprehensive Cancer Center Leukemia/Lymphoma Multidisciplinary Committee; OR their condition precludes waiting to search and find a donor in the National Marrow Donor Registry Exclusion Criteria: * Female patients who are pregnant or breast-feeding * HIV or HTLV-1 positivity * Any leukemia with a morphologic relapse or persistent disease in the BM with \>= 20% blasts (cytogenetic relapse without morphologic evidence of relapse, or cytogenetic persistent disease is acceptable) * Active extramedullary leukemia, including CNS disease * Prior hematopoietic stem cell transplant (autologous or allogeneic) * Uncontrolled infection * Patient has an identical related bone marrow donor or a 6/6 HLA-identical matched unrelated donor * Any patient who is unable to provide informed consent or comply with the requirements of the protocol