Decitabine in Treating Patients With Metastatic Papillary Thyroid Cancer or Follicular Thyroid Cancer Unresponsive to Iodine I 131

Inclusion Criteria: * Histologically confirmed papillary thyroid or follicular thyroid carcinoma: * Differentiated disease; * Metastatic disease documented by ultrasound, computed tomography (CT) scan (without iodinated contrast), or MRI - All metastatic disease foci =\< 10 mm in all dimensions * Must have been treated with total or near-total thyroidectomy AND at least 1 course of iodine I 131 (131I)(\>=29.9 mCi) OR demonstrated negative uptake on a postoperative low-dose131I scan * Must have undergone whole body 131I scan 1-3 days after administration of =\< 5.5 mCi of 131I demonstrating no visible iodine uptake within the lesions unless demonstrated negative uptake on a postoperative low-dose131I scan within the past 12 weeks: * Must have 24-hour urine iodine excretion =\< 500 mcg within 1 week of 131I scan * Must be receiving thyroid hormone therapy AND have thyroid-stimulating hormone level =\< 0.5 mU/L * No known brain metastases * Performance status: * Eastern Cooperative Oncology Group (ECOG) 0-2 OR Karnofsky 60-100% * Hematopoietic: * Absolute neutrophil count \>= 1,500/mm3; * Platelet count \>= 100,000/mm3; * White Blood Count (WBC) \>= 3,000/mm3 * Hepatic: * aspartate aminotransferase-alanine aminotransferase (AST and ALT) =\< 2.5 times upper limit of normal; * Bilirubin normal * Renal: * Creatinine not elevated OR * Creatinine clearance \>= 60 mL/min * Cardiovascular: * No symptomatic congestive heart failure; * No unstable angina pectoris; * No cardiac arrhythmia * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * No prior allergic reaction attributed to compounds of similar chemical or biological composition to decitabine * No concurrent uncontrolled illness * No active or ongoing infection * No psychiatric illness or social situation that would preclude study compliance * No prior cytotoxic chemotherapy for thyroid cancer * At least 6 months since prior external beam radiotherapy administered for locoregional disease in the thyroid bed or to the cervical or upper mediastinal lymph node regions (no more than 6,000 cGy) * More than 6 months since other prior radiotherapy and recovered * More than 6 months since prior therapeutic 131I \> 10 mCi * More than 18 months since prior cumulative 131I activity of at least 500 mCi * More than 12 months since prior amiodarone (Unless 24-hour urinary iodine excretion is =\< 500 mcg) * No concurrent combination antiretroviral therapy for HIV-positive patients * No other concurrent anticancer therapy * No other concurrent investigational agents * More than 6 months since prior intrathecal iodinated contrast (Unless 24-hour urinary iodine excretion is =\< 500 mcg) * More than 3 months since prior IV or oral iodinated contrast for radiographic studies (Unless 24-hour urinary iodine excretion is =\< 500 mcg)