Donor Umbilical Cord Blood Transplant in Treating Patients With Advanced Hematologic Cancer

DISEASE CHARACTERISTICS: * Diagnosis of 1 of the following advanced hematologic malignancies: * Acute myeloid leukemia (AML) meeting the following criteria: * Not expected to be curable with chemotherapy and meets ≥ 1 of the following criteria: * High-risk cytogenetics (-7, -7q, -5, -5q, t\[6,9\], t\[9,11\], complex, Philadelphia chromosome positive \[Ph+\]) * AML evolved from prior myelodysplasia * AML secondary to prior chemotherapy * Failed to achieve remission * In second or subsequent remission * Marrow blasts ≤ 10% (may be achieved using chemotherapy) * Myelodysplastic syndromes (MDS) with high-risk features * International Prognostic Scoring System (IPSS) score intermediate -2 or high-risk * Marrow blasts ≤ 20% (may be achieved using chemotherapy) * Acute lymphoblastic leukemia meeting the following criteria: * Not expected to be curable with chemotherapy and meets ≥ 1 of the following criteria: * High-risk cytogenetics (Ph+, t\[4,11\], 11q23 abnormalities, and monosomy 7) * Required \> 1 induction course to achieve remission * Failed to achieve remission * In second or subsequent remission * Marrow blasts ≤ 10% (may be achieved using chemotherapy) * Chronic myelogenous leukemia meeting ≥ 1 of the following criteria: * Accelerated phase * Chronic phase refractory to imatinib mesylate * Blastic phase * Marrow blasts ≤ 10% (may be achieved using chemotherapy) * Multiple myeloma meeting 1 of the following criteria: * Stage II or III disease with \> first relapse or refractory disease * Newly diagnosed disease with chromosome 13 abnormalities * Lymphoma meeting the following criteria: * One of the following subtypes: * Diffuse large cell lymphoma * Mantle cell lymphoma * Peripheral T-cell lymphoma * T-natural killer (NK) cell lymphoma * Hodgkin's lymphoma * Disease failed to respond to primary therapy, progressed, or recurred after prior therapy * Patients who have failed autologous stem cell transplantation are eligible provided it has been \> 1 year since transplant * No rapid progression of malignant disease * Not eligible for autologous stem cell transplantation * Available umbilical cord blood (1-3 units) donor matching at ≥ 4 of 6 HLA antigens (A, B, and DR) * Patients with an HLA-identical or 1 antigen-mismatched related donor OR a potential HLA-matched unrelated donor matching at \> 6/8 (A, B, C, DR) alleles are not eligible PATIENT CHARACTERISTICS: * ECOG performance status 0-2 * Creatinine \< 2.0 mg/dL * Creatinine clearance \> 40 mL/min * Bilirubin \< 2.0 mg/dL * AST and alkaline phosphatase \< 3 times upper limit of normal * Hepatitis C and active hepatitis B allowed if patient has ≤ grade 2 inflammation or fibrosis by liver biopsy * Ejection fraction \> 40% by echocardiogram or MUGA * DLCO \> 40% of predicted * Not pregnant or nursing * Negative pregnancy test * No known HIV infection * No active infection requiring ongoing antibiotic treatment PRIOR CONCURRENT THERAPY: * See Disease Characteristics