DISEASE CHARACTERISTICS: Any of the following hematologic malignancies: Acute myelogenous leukemia in second or greater remission or relapse High risk acute lymphoblastic leukemia First complete remission with high risk cytogenetics [e.g., t(9;22), t(8;14), or t(4;11)] or slow early response OR Any second or greater remission OR Relapse or induction failure Relapsed non-Hodgkin's lymphoma (NHL) with chemotherapy sensitive disease Multiple myeloma with chemotherapy sensitive disease Chronic lymphocytic leukemia with chemotherapy sensitive disease Myelodysplastic syndrome Chronic myelogenous leukemia (CML) in accelerated phase, blast phase, or remission after blast phase Juvenile CML Patients must have haploidentical donors and meet the following criteria: Lack an HLA A, B, and DR matched related donor Lack an HLA A, B, DRB1 matched or single allele mismatched unrelated donor or for whom the following apply: High likelihood of rapid disease progression while an unrelated donor search is in progress AND Review of the patient HLA typing, gene frequency, results of the preliminary search and the donor pool suggest that a donor will not be found within a suitable time frame Lack an HLA A, B, DRB1 matched cord blood unit or lack a cord blood unit that has an adequate cell dose and is mismatched for 1-2 HLA A, B, DRB1 Do not have an indication for autologous transplant, including patients with NHL or multiple myeloma who are not eligible for autologous transplant Negative antidonor antibody crossmatch