LDS apologist Michael R. Ash has a very interesting argument about why there are no remains of horses found in the Americas which can be dated to the Book of Mormon “historical” time period. Ash says (complete comments here):

In the 4th & 5th centuries AD, the Huns of Central Asia and Eastern Europe had so many horses that estimates suggest that each warrior may have had up to ten horses. Horses were the basis of their wealth and military power. According to a non-LDS leading authority on the zoological record for central Asia, however, we know very little of the Huns' horses, and not a single usable horse bone has been found in the territory of the whole empire of the Huns.Based on the fact that other–once thriving–animals have disappeared (often with very little trace), it is not unreasonable to suggest that the same thing might have happened with the Nephite “horse.”

However, Ash is mistaken that there are no evidences for Hun horses. In 1990 archaeologists excavated the intact tomb of a Hun princess in Mongolia. Inside her coffin, according to the publication Mongolia Today:

The wooden box inside which the coffin was placed, contained a bronze jar with great engravings of various animals, metal bridle, details of horse equipment. A chariot wheel attracted special attention of archeologists as it has more than 40 holes for rungs and therefore the wheel must be very large.

Five horse skulls were put on the northern side to the burial, with one horse head turned towards the coffin. The number 5 was revered by Huns because of their special reverence for Cygnus Constellation. One separate horse head probably belonged to the princess’ beloved horse.

Notice how carefully Ash worded his argument: “not a single useable horse bone.” (I'm not sure what makes an ancient bone “useable,” but he put the emphasis on the bones.) While the Hun princess's grave was remarkable because it contained one of the very few Hun horse skeletal remains, it also contained something much more common: horse tack such as a bridle and other equipment.

Here's the conclusion: It is possible for something such as fragile and decomposable as horse bones to nearly completely disappear after more than a thousand years. However, it is nearly impossible for all the equipment –typically of metal and leather — associated with great herds of horses to disappear as well. And I know of no horse tack found in the Americas that dates to before Columbus.

For more information, see The Mormon Mirage 3rd Edition:  A Former Member Looks at the Mormon Church Today (Zondervan, 2009). Also available as an audiobook and as an expanded-text E-book for Nook, Kindle and other reading devices.