This phase II trial studies how well pembrolizumab works when given together with ixazomib citrate and dexamethasone in treating patients with multiple myeloma that has come back (relapsed). Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Ixazomib citrate may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Drugs used in chemotherapy, such as dexamethasone, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving pembrolizumab together with ixazomib citrate and dexamethasone may work better in treating patients with multiple myeloma.