We certainly do! It can be difficult to fathom how a peoples can retain their histories without written words, but the indigenous peoples of North America did so for much of their history, until colonialism reared its ugly head.

First, you must understand that like your examples of Europe and Asia, there are many different nations within indigenous North America. These nations will have a unique history, culture and beliefs. Some were sedentary cultures, remaining in one place in longhouses or wigwams, others were nomadic, living in the infamous tipis or igloos. The dynamics of their family situations varied, some were matrilineal (the Haudenosaunee being the most famous example), some Pacific northwest nations had very strict class systems, others were very loosely organized (mainly the nomadic nations). These are just some examples of pre-contact history, although many things have been lost. There are of course many many different creation stories and a wonderful mythology for the different nations.

Secondly, one must take into account that much of indigenous history was passed down orally (although some did have physical records, such as the infamous totem pole) and much of these traditions were forced out of the indigenous peoples by colonialists. My area of study is indigenous peoples of Canada so I will focus on that. First, contact by Europeans caused close to 95% of indigenous peoples to die due to disease early in post-contact history. Indigenous peoples then were subjected to forced assimilation, any gatherings were banned and their children were removed and placed into Residential Schools. These schools were, to quote Duncan Campbell Scott, to "take care of the Indian problem." The children were beaten for displaying any of their cultures, beliefs or speaking their languages (they were also subjected to sexual abuse and such poor living conditions that resulted in mass burials). While the last residential school was closed in 1996, forcing the children into the schools began to fall out of favour sometime in the 1950s. This then led to the "60s scoop" where many indigenous children were removed from the "Indian" homes and placed in white Canadian homes. One may argue that this continues on to this day, as indigenous children continue to be overrepresented in Canadian welfare system. So much that they have to house some of these kids in motels. These things have led to some loss of history, but with the help of anthropologists and linguists, the histories are being slowly pieced together.