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