NK Cells in Cord Blood Transplantation

Inclusion Criteria: * Patients must have one of the following hematologic malignancies: acute myelogenous leukemia (AML), induction failure, high-risk for relapse first remission (with intermediate-risk or high-risk cytogenetics, fms-related tyrosine kinase 3 \[FLT3\] mutation positive and/or evidence of minimal residual disease by flow cytometry), secondary leukemia from prior chemotherapy and/or arising from myelodysplastic syndrome, any disease beyond first remission * Myelodysplastic syndrome (MDS): primary or therapy related * Acute lymphoblastic leukemia (ALL): induction failure, primary refractory to treatment (do not achieve complete remission after first course of therapy) or are beyond first remission including second or greater remission or active disease; patients in first remission are eligible if they are considered high risk, defined as any of the following detected at any time: with t(9;22) or t(4;11), hypodiploidy, complex karyotype, secondary leukemia developing after cytotoxic drug exposure, and/or evidence of minimal residual disease, or acute biphenotypic leukemia, or double hit non-Hodgkin's lymphoma * Non-Hodgkin's lymphoma (NHL) in second or third complete remission, refractory NHL, or relapsed NHL (including relapse post autologous hematopoietic stem cell transplant); double hit lymphomas in first remission or more advanced disease * Small lymphocytic lymphoma (SLL), or chronic lymphocytic leukemia (CLL) with progressive disease following standard therapy; patients with progressive CLL following standard therapy who meet European Bone Marrow Transplant (EBMT) consensus guidelines of indications for allogeneic stem cell transplantation; this includes patients with (1) lack of response or early relapse within 1 year of receiving a purine analog-containing treatment regimen, (2) disease relapse within 2 years of receiving a purine analog combination therapy or after other therapies such as autologous stem cell transplantation, and (3) CLL associated with tumor protein (p)53 mutations or deletions and/or deletion (del) (17p) requiring therapy; patients must have chemosensitive disease with at least a partial response (PR) or stable disease (SD) with last treatment regimen * Chronic myeloid leukemia (CML) second chronic phase or accelerated phase * Hodgkin's disease (HD): induction failures, second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant) * Multiple myeloma: stage II or III, symptomatic, secretory multiple myeloma requiring treatment * Performance score of at least 80% by Karnofsky or 0 to 2 Eastern Cooperative Oncology Group (ECOG) * Left ventricular ejection fraction greater than 45% * Pulmonary function test (PFT) demonstrating a diffusion capacity of least 45% predicted * Creatinine \< 1.6 mg/dL * Serum glutamate pyruvate transaminase (SGPT) =\< to 2.0 x normal * Bilirubin =\< to 2.0 x normal * All study participants must be registered into the mandatory Revlimid Risk Evaluation and Mitigation Strategy (REMS) program, and be willing and able to comply with the requirements of the Revlimid REMS program * Females of reproductive potential must adhere to the scheduled pregnancy testing as required in the Revlimid REMS program * Negative beta human chorionic gonadotropin (HCG) test in a woman with child bearing potential defined as not post-menopausal for 24 months or no previous surgical sterilization and willing to ongoing pregnancy testing while on treatment with lenalidomide * Woman with child bearing potential must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method AT THE SAME TIME, at least 4 weeks before she starts taking lenalidomide * Men must agree to use a latex condom during sexual contact with females of child bearing potential even if they have had a successful vasectomy * Patients must have two CB units available which are matched with the patient at 4, 5, or 6/6 human leukocyte antigen (HLA) class I (serological) and II (molecular) antigens; each cord must contain at least 1.5 x 10\^7 total nucleated cells/Kg recipient body weight (pre-thaw); cord blood units will be procured through the National Marrow Donor Program (NMDP) * Have identified a backup cells source in case of engraftment failure; the source can be autologous, related or unrelated * Patients who have had a prior autologous transplant are eligible Exclusion Criteria: * Patients with known history of human immunodeficiency virus (HIV)/acquired immune deficiency syndrome (AIDS) * Active central nervous system (CNS) disease in patient with history of CNS malignancy * Patients with chronic active hepatitis or cirrhosis; if positive hepatitis serology, the study chair may deem the patient eligible based on the results of liver biopsy * Patients with known hypersensitivity to lenalidomide and/or rituximab * Patients who have a matched related donor who is eligible and willing to donate stem cells