Here's some places that might fit the bill..... Most probably do not just serve Native dishes, but many items are local and inspired by Native Americans.
In Santa Fe we like Amaya Restaurant at Hotel Santa Fe which is owned by Picuris Pueblo. Ronald Roybal plays there, Indian Flute and Spanish Guitar.
Tamaya Resort on Santa Ana Pueblo north of Bernalillo has a great restaurant, Corn Maiden.
I have also heard decent reviews of O The Eating House in Pojoaque, owned by the pueblo.
A place off in AZ is the restaurant at La Posada in Winslow AZ, The Turquoise Room. The chef there tries to purchase locally grown/raised food items. We loved this place last time we were through there.
Farview Lodge inside Mesa Verde National Park in southern Colorado has the Metate Room.. enjoyable dining in a spectacular location.
Another time to taste native foods is during special events..
Gallup Inter-tribal Ceremonial
Gathering of Nations in Albuquerque
Santa Fe Indian Market
Zuni Harvest Festival
Pueblo Feast days