Laboratory-Treated T Cells in Treating Patients With High-Risk Relapsed Acute Myeloid Leukemia, Myelodysplastic Syndrome, or Chronic Myelogenous Leukemia Previously Treated With Donor Stem Cell Transplant

Inclusion Criteria: * Patients must express HLA-A\*0201 * Patients who are currently undergoing or who previously underwent matched allogeneic HCT for: * AML: Prospective enrollment will now be limited to patients with relapsed disease (overt relapse or minimal residual disease) at any time post allogeneic HCT * MDS will no longer be a criterion for eligibility * CML will no longer be a criterion for eligibility * Patients must have an HLA-matched donor of hematopoietic stem cells (related or unrelated) * Patients must be able to provide blood and bone marrow samples and undergo the procedures required for this protocol * Patients must be \>= 15 kg, as patients with lower weight would be incapable of providing high volume and frequent blood samples for monitoring and analysis * Patients must be able to give informed consent; parent or legal representative will be asked to consent for patients younger than 18 year old * DONOR: Patient and donor (related or unrelated) must be HLA-matched and express HLA-A\*0201 * DONOR: Donor must be Epstein-Barr virus (EBV) or cytomegalovirus (CMV) seropositive * DONOR: Donor must be age 18 or older * DONOR: In good general health * DONOR: Able to give informed consent Exclusion Criteria: * Central nervous system (CNS) tumor refractory to intrathecal chemotherapy and/or cranio-spinal radiation * In patients whose leukemic cells are available for evaluation, the expression of WT1 in the patient's bone marrow will be determined; if WT1 expression in the patient's bone marrow is not highly expressed by polymerase chain reaction (PCR), the patient will be excluded from the study; patients with no evaluable leukemia will be eligible for enrollment based on the high frequency of positive leukemias (\> 90%), and leukemia will be evaluated for WT1 expression if recurrence is detected * Human immunodeficiency virus (HIV) seropositive; testing for HIV should be within 6 months of enrollment * Medical or psychological conditions that would make the patient unsuitable candidate for cell therapy at the discretion of the principal investigator (PI) * Pregnancy or breast-feeding; women of childbearing potential must have a negative serum or urine beta-human chorionic gonadotropin (B-hCG) pregnancy test result within 14 days before the first dose of WT1-specific T cell infusion; woman of non-childbearing potential will be defined as being postmenopausal greater than one year or who have had a bilateral tubal ligation or hysterectomy; all recipients of WT1-specific T cells will be counseled to use effective birth control during participation in this study and for 12 months after the last T cell infusion * DONOR: Less than 18 years old * DONOR: Active infectious hepatitis * DONOR: HIV or human T-lymphotropic virus (HTLV) seropositive * DONOR: Pregnancy or nursing * DONOR: Significant medical conditions (e.g. immunosuppressive therapy) that would make the donor an unsuitable T cell donor * DONOR: Unable to give informed consent