I. Determine the response rate of cediranib in combination with lenalidomide in patients with iodine refractory, unresectable DTC who have evidence of disease progression within 12 months of study enrollment. (Phase I) II. Determine the toxicity, duration of response, progression free survival, and overall survival in patients with DTC treated with cediranib plus lenalidomide. (Phase I) III. Determine response rates and duration of response, early tumor size changes, the toxicity, and overall survival in patients with DTC treated with cediranib or cediranib plus lenalidomide. (Phase II) IV. Determine whether the presence of v-raf murine sarcoma viral oncogene homolog B1 (B-RAF) or V-Ki-ras2 Kirsten rat sarcoma (K-RAS) mutations in patients with DTC predict response to cediranib or cediranib plus lenalidomide. (Phase II)