The purpose of the study is to determine if AMG 706 will have clinically meaningful anti-tumor activity in subjects with locally advanced or metastatic thyroid cancer who are not candidates for radioactive iodine therapy or local therapies.

Inclusion Criteria: - Histologically confirmed locally advanced or metastatic thyroid cancer - Measurable disease - Normal blood pressure (if history of hypertension, blood pressure must be controlled with medication) - Evidence of disease progression within 6 months before starting study (for differentiated thyroid cancer subjects) - Evidence of disease progression within 6 months before starting study OR symptomatic disease (for medullary thyroid cancer subjects) - Not a candidate for surgical resection, external beam radiotherapy, radioiodine therapy, or other local therapy - At least 18 years of age Exclusion Criteria: - Undifferentiated/anaplastic thyroid cancer - Untreated or symptomatic brain metastases - Prior malignancy, unless cured with treatment and no evidence of disease for greater than or equal to 3 years before starting study (history of thyroid cancer, in situ cervical cancer, or basal cell cancer of skin are exceptions) - Myocardial infarction or any unstable cardiac condition (e.g., congestive heart failure, unstable angina) within 1 year before starting study - Arterial thrombosis or deep vein thrombosis within 1 year before starting study