Most tribes do have some form of carving.  Whether in the form of a battle club, effigy, or simply for decoration, carvings were more than likely made possible because of access to wood or other organic material (bone, baleen, etc.).  Even in the Southwest, willow and cottonwood trees provided a source of energy (fire) and a functional raw material for creating carvings.

Carvings fulfill many roles.  Whether it is ceremonial in nature, a tool, a mask, or a weapon, the workmanship has become more than an effort, it has become an artistic statement.  While it is certainly true that the 'older' carvings demand more at auctions, the newer carvings are just as beautiful. 

Hopi Carvings
The Hopi have created some of the most beautiful carvings.  Whether in the form of a tablita or whether in the form of a kachinas (sometimes called katsinas), a polychromatic deity representation plays a large part in their ceremonies.
Each 'carving' represents a deity.  Each will have a unique look, but an overall 'theme'.  They are sometimes given to children during ceremonies.   

Inuit Carvings
The soapstone carvings of the Inuit continue to be an important part of their culture.  They are also known for their scrimshaw, totem poles, bowls, masks, and rattles.  More information is coming on this subject.