Romidepsin in Treating Patients With Recurrent and/or Metastatic Thyroid Cancer That Has Not Responded to Radioactive Iodine

Inclusion Criteria: * Histologically or cytologically confirmed non-medullary thyroid carcinoma, including the following cell types: * Papillary * Follicular * Variants of papillary or follicular * Hürthle cell * Recurrent and/or metastatic disease * Measurable disease * At least 1 unidimensionally measurable lesion \>= 20 mm by conventional techniques OR \>= 10 mm by spiral CT scan * Progressive disease during or after prior treatment, as defined by \>= 1 of the following criteria: * Presence of new or progressive lesions on CT scan or MRI * New lesions on bone or positron-emission tomography scan * Rising thyroglobulin level * Minimum of 3 consecutive rises with an interval of \>= 1 week between each determination * Refractory to radioactive iodine (RAI) * Absent or insufficient RAI-uptake documented by whole-body RAI scan within the past 6 months * At least 1 lesion with absent RAI-uptake required for insufficient uptake * No known brain metastases * Performance status - Karnofsky 60-100% * WBC \>= 3,000/mm\^3 * Absolute neutrophil count \>= 1,500/mm\^3 * Platelet count \>= 100,00/mm\^3 * Bilirubin normal * AST and ALT =\< 2.5 times upper limit of normal * Chronic active viral hepatitis allowed provided patient is clinically stable and fulfills liver function eligibility criteria * Creatinine normal * Creatinine clearance \>= 60 mL/min * QTc =\< 480 msec by ECG * ST segment depression \< 2 mm * LVEF \>= 50 % by echocardiogram * No left ventricular hypertrophy, as defined by end-diastolic wall thickness \> 12 mm in both the left ventricular posterior wall as well as septum or restrictive cardiomyopathy * No history of any of the following cardiac diseases: * Canadian Cardiovascular Society (CCS) class II-IV angina pectoris * Myocardial infarction within the past 12 months * Sustained ventricular tachycardia, ventricular fibrillation, Torsade de Pointes, or cardiac arrest unless currently addressed with an automatic implantable cardioverter defibrillator * Any cardiac arrhythmia requiring digitalis or another antiarrhythmic medication other than a beta blocker or calcium channel blocker * No uncontrolled hypertension (i.e., blood pressure \>= 160/95) * Mobitz II second degree block in patients who do not have a pacemaker * First degree or Mobitz I second degree block, bradyarrhythmias or sick sinus syndrome require Holter monitoring and evaluation by cardiology * Uncontrolled dysrhythmias * No history of congenital long QT syndrome * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * Thyroid stimulating hormone normal or suppressed * No history of allergic reaction attributed to compounds of similar chemical or biologic composition to FR901228 * No ongoing or active infection * No psychiatric illness or social situation that would preclude study participation * No other concurrent uncontrolled illness * At least 4 weeks since prior biologic or targeted agents (e.g., interferon alfa, thalidomide, octreotide, or cetuximab) * No concurrent antineoplastic biologic agents * No prior FR901228 (depsipeptide) * No prior cytotoxic chemotherapy * Cytotoxic chemotherapy as a radiosensitizer allowed provided \>= 3 months since prior administration * No other concurrent antineoplastic chemotherapy * Not specified * At least 4 weeks since prior external beam radiation therapy * Documented disease progression required if patient received external beam radiotherapy to index lesions * At least 3 months since prior RAI therapy * Diagnostic studies using =\< 12 mCi of RAI are not considered RAI therapy * No concurrent antineoplastic radiotherapy * At least 2 weeks since prior anticancer cyclooxygenase-2 (COX-2) inhibitors, isotretinoin, or complementary medications * At least 4 weeks since prior tyrosine kinase inhibitors (e.g., gefitinib or erlotinib) * No other concurrent investigational agents * No other concurrent anticancer therapy * No concurrent drugs known to have histone deacetylase inhibitor activity (e.g., valproic acid) * No concurrent combination anti-retroviral therapy for HIV-positive patients * No concurrent hydrochlorothiazide * No concurrent treatment dose warfarin * No concurrent agents that cause QTc prolongation * Concurrent daily aspirin given after myocardial infarction or COX-2 inhibitors at standard anti-inflammatory or pain doses allowed