This clinical trial is for subjects with multiple myeloma that has returned after treatment (relapsed) or did not respond to prior treatment (refractory). The study is in two parts, Phase I and Phase II. Phase I will determine the maximum tolerated dose of romidepsin in combination with pomalidomide and dexamethasone. The purpose of Phase II is to evaluate the effectiveness of combining romidepsin with pomalidomide and dexamethasone. The hypothesis is that overall response in a cohort of patients treated with romidepsin + pomalidomide + dexamethasone will be 60 percent.