Mryntu

The Maya are known for their astronomy and mathematics, most notably their famous calculation of the length of a year. I was surprised to learn that they supported a flat Earth model, though to be fair, a spherical Earth wasn't yet fully accepted in the Old World, either.

Do we know if other pre-Columbian societies knew the true shape of the Earth? It doesn't look like there's much research on this topic, which is maybe not surprising given the lack of written records. I'm interested in knowing what oral tradition says about this. In any case, you don't need advanced technology to discover that the Earth is a sphere, and the conclusion may have been reached independently many times in Eurasia.

Firstly, it is important to be aware that our understanding of the civilisations and culture of Pre-Columbian America is far from complete. Secondly, I don't propose to attempt to cover every civilisation, so this will - at best - be only a partial answer.

However, with those caveats:

As far as I am aware, we have no evidence that any pre-Columbian cultures or civilisations knew the true shape of the Earth. All the evidence we do have supports the idea that they all supported a flat-Earth model. What is more, there seems to be a considerable degree of overlap in the cosmologies of the major civilisations.

Aztec:

The Aztecs believed that the Earth had the approximate shape of a giant disk. That disk was divided into four cardinal directions. Above the Earth were 13 heavens, while below the Earth were the 'nine Hells of Mictlan'.

• Encyclopedia Britannica 'Pre-Columbian civilizations'


• Miguel León Portilla: Aztec Thought and Culture: A Study of the Ancient Nahuatl Mind
  

Maya:

The Mayans also believed that 13 heavens were stacked in layers above the Earth. The Earth itself was flat and four-cornered, and carried on the back of a giant reptile (probably a crocodile) which was, in turn,floating on the ocean. Below the Earth were nine underworlds once again.

• Encyclopedia Britannica 'Pre-Columbian civilizations'


• Canadian Museum of History

Inca:

The Inca called their land "Tahuantinsuyu" which (according to the course notes for Lecture 4 of Astronomy 161 at Ohio State) means "The Four Quarters of the Earth". This suggests that they had similar views to those of the Aztec and the Maya. In his paper Animals and Astronomy in the Quechua Universe, Gary Urton observed:

"... the celestial River is believed to carry into the sky the actual water which flows through the Vilcanota River. As the Vilcanota River flows from the southeast to the northwest, it carries terrestrial water to the edge of the earth. The water then flows into the mar, the cosmic sea, which completely encircles the earth.

• p 111 (my emphasis)

Again, this suggests that the Inca also believed that the Earth was flat and four-cornered.