This phase I trial tests the safety, side effects, and best dose of gamma-primed mesenchymal stromal cells in preventing acute graft versus host disease after donor stem cell transplant in patients with blood cancers or myelodysplastic syndrome. Acute graft versus host disease (AGVHD) is a condition that might occur after a donor stem cell transplant. AGVHD happens when certain white blood cells, called T-cells, in the donor cells (the graft) attack the patient's body. AGVHD causes skin, liver, stomach and intestinal problems. Mild AGVHD is often not difficult to treat. However, AGVHD can be more severe (blistering skin rash, severe or bloody diarrhea and severe jaundice). Severe cases are more difficult to treat and can be fatal. Mesenchymal stromal cells (MSCs) are cells that exist in tissues throughout the body. One place they are found is in the bone marrow. They can be obtained by needle aspiration; the same way bone marrow samples are obtained to test for leukemia. MSCs can suppress immune cells—like the ones that cause GVHD—and can promote healing. In this study, MSCs are exposed to a substance that is made by cells of the immune system (cytokine) called interferon gamma, which will activate them before they are infused. Gamma-primed mesenchymal stromal cells may work better than standard medications in preventing AGVHD.