This phase I trial studies the side effects and best dose of dinaciclib and bortezomib when given together with dexamethasone in treating patients with multiple myeloma that has returned after a period of improvement. Dinaciclib and bortezomib may stop the growth of cancer 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 dinaciclib and bortezomib together with dexamethasone may kill more cancer cells.