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