This phase II trial studies how well panobinostat, carfilzomib, and dexamethasone work in treating patients with multiple myeloma that has come back or does not respond to treatment. Panobinostat may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Drugs used in chemotherapy, such as carfilzomib and 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. Using multiple myeloma cells from patients' blood samples, the researchers will do laboratory tests to look at how well each of the drugs, alone and in different combinations, kill multiple myeloma cells. If the laboratory tests work well, they may be used in the future to help plan treatment for future patients.