I would argue that a health care issue that is common to many different native tribes in the United States today is a lack of access to adequate medical care. Poverty is often the leading indicator and factor affecting health care access, and while not all tribes could be classified as poor, Census data suggests they are disproportionately so overall.
Native reservations in general, but specifically in New Mexico, Texas, Arizona, Montana and Alaska, tend to be relatively isolated from population centers where both primary and more advanced care are readily available.
There are some signs that conditions are improving. Revenue from tribal casinos has, in some instances, led to an infusion of money into tribal health clinics and outreach programs. National health care reform efforts, if successful, could improve access as well.