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 vent…