Except that the northern plains tribes do talk about the coming of the horse in their tribal oral histories. It's what allowed the Sioux, for instance, to move out of the Minnesota woods and onto the Great Plains in the early 18th Century, and to move from small shelters into bigger teepees that they could only transport with the help of horses and travois.
Archaeology shows no signs of pre-Columbian Indians using horses. No petroglyphs. No artifiacts. No horse bones have been found dating between the extinction ca. 10,000 years ago and ca. 1500 AD. Furthermore, Indians wouldn't need to just spontaneously figure out that you could ride these new animals. The Mexican Indians saw the Spanish riding and picked it up. Then it was just a matter of the next tribe north seeing that it could be done while the horse population exploded across the plains with ample food and no predators to speak of. Two hundred years is plenty of time for that knowledge to spread from Mexico to Minnesota.
Thanks.
Trade carried various things like copper and unusual stones, and those are found in contexts that show just that. The lack of perishable items in old sites is likely due to their decomposition. Same goes for bones. Human remains are especially scarce in the Americas, so it shouldn't be a surprise that horse remains are.
When horse bones (or petroglyphs) have been found, they have been used as diagnostic of post-Columbian date for the other remains -- but that's not a sound idea, based as it is on the assumption that horses were reintroduced.
Anyway, here's some more folklore, which purports to show (or can easily be purported to show) conclusively that the tribal cultures were altered by Euro-meddling:
Spanish Colonial Horse and the Plains Indian Culture
by O. Ned Eddins
http://www.thefurtrapper.com/indian_horse.htm
...Francis Haines states that by the early seventeen hundreds all the tribes south of the Platte had some familiarity with horses. Pierre Gaultier de La Verendrye a French trader reached the Mandan village on the Missouri River in 1738, while there he heard of Indians to the south that had a few horses. George Hyde estimated that 1760 was the period the Teton Sioux acquired horses from Arikara. In 1768, Jonathan Carver found no horses among the Dakota Sioux of upper Missouri, but two years later the Yankton Sioux had horses... It was the Spanish horse that made it possible for the American Indians to move onto the Plains and become truly nomadic. The individual, not the tribe, owned the horses. This produced a class system based on ownership of horses -- those with and those without. Horses spread through the Arikara to the Missouri River villages of the Mandan and Hidatsa and eventually to the Sioux and the Cheyenne. When the first white traders reached the Plains none of the Indians North and East of the Black Hills had horses. By the end of the seventeen hundreds, the Indian horse had reached most of the Rocky Mountains and Plains Indians. An extensive Indian trade network existed between the Indian tribes decades before explorers and fur traders reached the Missouri River villages.
Introduction to the Ute Tribal History
http://www.southern-ute.nsn.us/history/intro.html
Possession of horses allowed the Utes to begin buffalo hunting on the eastern slope of the Rocky Mountains and the buffalo soon became one of their main resources, because it would provide them with many useful products: e.g., meat for food (one wouldn't have to work so hard gathering food); hides for tipi covers, blankets, clothing, moccasins, and bags of all kinds; sinew thread for sewing and for bowstrings; horn and hoof glue for many purposes. And with the horse, the Utes could more easily evade their enemies, transport their goods to a central camp where the women and children were protected, and range farther to hunt for food.
The impact of wild horses on the Native American people
http://www.indians.org/articles/wild-horses.html
For the Native Americans who revered the wild horses, they took great care in capturing and training them for practical uses, always mindful of the horses role in the spirit world. With the introduction of the wild horses, their culture and way of life broaden dramatically. Now, the Native Indians could hunt for buffalo and other food more easily. They could trade and barter with other tribes and even increase their claim of land.
The Coso Painted Style (Pictographs)
By Alan Garfinkel Ph.C.
http://www.petroglyphs.us/article_the_Coso_Painted_Style_pictographs.htm
An exceptionally harsh winter of 1861-1862 led to potential starvation by the Native peoples of eastern California. Much of the area had already been deforested to supply timber and charcoal for the mines. The timbers were harvested from the substantial stands of pinyon trees that would normally have provided significant nut crops in the fall. With their traditional subsistence practices in disarray, Native peoples began raiding Euroamerican cattle and horses. It was during this period that depredations began and Native peoples murdered White settlers in considerable numbers.
Petroglyphs of a horse (above) and man (below) at Picture Canyon
http://www.ou.edu/okage/lodgepole2/Day4Petroglyph.jpg
http://www.ou.edu/okage/lodgepole2/journal4.html
Since the games pre-dated the Spanish, what should we suppose the Mayans were riding?