Sunitinib Malate in Treating Patients With Thyroid Cancer That Did Not Respond to Iodine I 131 and Cannot Be Removed by Surgery

Inclusion Criteria: * Patients must have histologically or cytologically confirmed papillary, follicular, or Hurthle cell carcinoma (cohort A) or medullary thyroid carcinoma (cohort B); their disease must have progressed despite treatment with iodine-131 therapy or they are not candidates for iodine-131 therapy and their disease cannot be completely removed by surgery; all patients with WDTC are expected to be on thyroxine suppression therapy * Patients must have radiographically or biochemically measurable disease; radiographically measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan; biochemically measurable disease is defined as an elevated thyroglobulin (WDTC patients) or calcitonin (MTC patients) * Patients must have evidence of disease progression (objective growth of existing tumors or rising thyroglobulin or calcitonin levels) within the last 6 months * Patients cannot have received prior receptor tyrosine kinase inhibitors; patients cannot have received more than one prior chemotherapy regimen for metastatic disease; patients cannot have received prior external beam radiation to the measured tumor constituting the target lesion(s) * Life expectancy of greater than 12 weeks * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 (Karnofsky \>= 60%) * Leukocytes \>= 3,…