MYTH: The ancient Jaredites used dinosaurs as beasts of burden.
It’s not really clear where this myth came from, but I have seen it more than once on the internet jokingly referred to as the “Flintstonian Theory.” This myth stems from the Book of Mormon’s reference to Cureloms, Cumoms and Elephants.
In the book of Ether, it states:
And they also had horses, and asses, and there were elephants and cureloms and cumoms; all of which were useful unto man, and more especially the elephants and cureloms and cumoms.
Cureloms and Cumoms are untranslated Jaredite terms. In other words, no description of these animals is ever given other than to say that they were used by man. John L. Sorenson suggests that they could be alpacas, llamas, mastodons or some other Pleistocene mammals. Sadly, these beasts are only referenced once in the Book of Mormon, and without further description we cannot know for sure what kind of animal they were. But, considering that the dinosaurs died out tens of thousands of years ago, it is probably pretty safe to say that cureloms and cumoms were not dinosaurs.
Complicating this myth however, is the Book of Mormon’s mention of elephants. Some LDS scholars have suggested that the word elephant actually referred to a mastodon, mammoth or some other, now extinct, variation of the elephant. However, Book of Mormon critics argue that both the mammoth and the mastodon died out thousands of years ago, long before the timing of the Jaredite’s mention of elephants (the time period of the elephant reference is estimated at roughly 2500 B.C.)
Certainly a reference claiming to have domesticated and used an extinct animal such as a mastodon or mammoth could loosely give rise to the idea of dinosaurs. And although most archeologists agree that elephantine animals lived in the America’s as some point in the continent’s past, such a claim by the Book of Mormon would be ludicrous if all forms of elephant actually were extinct during the Book of Mormon time period.
But were they?
According to Fairmormon.org, at least five elephant effigies have been found in ancient Mexico, indicating that the elephant and man co-existed. The website states:
Dr. Verrill, a well-known (non-Mormon) archaeologist describes one of these figures as “so strikingly and obviously elephantine that it cannot be explained away by any of the ordinary theories of being a conventionalized or exaggerated tapir, ant-eater or macaw. Not only does this figure show a trunk, but in addition it has the big leaf-like ears and the forward-bending knees peculiar to the elephants. Moreover, it shows a load or burden strapped upon its back. It is inconceivable that any man could have imagined a creature with the flapping ears and peculiar hind knees of an elephant, or that any human being could have conventionalized a tapir to this extent”... 
Additionally, butchered mastodon bones have been discovered dating to shortly after the time of Christ, and one site, dating to approximately 100 B.C., “has yielded the remains of a mammoth, a mastodon, as well as a horse.”
Some scholars have suggested that elephantine mammals lived as late at 2000 BC, putting them well within the time frame of the Jaredite era. In an article entitled “Men and Elephants in America,” Ludwell Johnson wrote:
Discoveries of associations of human and proboscidean remains [Elephantine mammals, including, elephants, mammoths, and mastodons] are by no means uncommon. As of 1950, MacCowan listed no less than twenty-seven” including, as noted by Hugo Gross, a “partly burned mastodon skeleton and numerous potsherds at Alangasi, Ecuador...There can no longer be any doubt that man and elephant coexisted in America.... Probably it is safe to say that American Proboscidea have been extinct for a minimum of 3000 years. 
Another elephant-like creature known as the Gomphotheres lived in the Americas from 12 million years ago until around 400 A.D.
The Gomphotheres are a diverse group of extinct elephant-like animals (proboscideans) that were widespread in North America during the Miocene and Pliocene epochs, 12-1.6 million years ago. Some also lived in parts of Eurasia and Beringia, and following the Great American Interchange, in South America. From about 5 million years ago onwards, they were slowly replaced by modern elephants, but the last South American species did not finally become extinct until as recently as 400 CE.
Gomphothere remains are common at South American Paleo-indian sites. One example is the early human settlement at Monte Verde, Chile, dating to approximately 14,000 years ago.
Gomphotheres differed from elephants in their tooth structure, particularly the chewing surfaces on the molar teeth. Most had four tusks, and their retracted facial and nasal bones prompt paleontologists to believe that gomphotheres had elephant-like trunks. 
It appears that archaeological evidence suggests there elephantine animals did in fact live in ancient America during the same time frame as the Jaredites, and possibly the Nephites as well.
Certainly there is no evidence, in fact it seems ludicrous to contend that actual dinosaurs existed during the Book of Mormon time period, but in the form of extinct elephants and undescribed/unknown animals such as the cureloms and cumoms, it would appear that we have found our mythical dinosaurs.
 The Book of Mormon, Ether 9:19
 John L. Sorenson, “An Ancient American Setting for the Book of Mormon” Salt Lake City, Utah: Deseret Book Company, Provo, Utah; Foundation for Ancient Research and Mormon Studies, 1996, 298
 A Mastodon is “a large extinct mammal that resembled an elephant, with shaggy hair and two sets of tusks.” Encarta Dictionary
 A mammoth is “a large extinct elephant that had long curved tusks and was covered with hair. It existed mainly in the northern hemisphere and died out more than 10,000 years ago.” Encarta Dictionary
 www.Fairmormon.org. “Book of Mormon anachronisms/Animals,” under the subheading “Elephant.” Retrieved February 27, 2009.
 www.Fairmormon.org. “Book of Mormon anachronisms/Animals,” under the subheading “Elephant.” Retrieved February 27, 2009. Referencing John L. Sorenson, An Ancient American Setting for the Book of Mormon (Salt Lake City, Utah : Deseret Book Co. ; Provo, Utah : Foundation for Ancient Research and Mormon Studies, 1996 ), 297–298.
 Ludwell Johnson. “Men and Elephants in America” published in Scientific Monthly, Oct. 1952
 "Gomphothere wikipedia.org (accessed 25 February 2009). The article is citing Palmer, D., ed. (1999). The Marshall Illustrated Encyclopedia of Dinosaurs and Prehistoric Animals. London: Marshall Editions. pp.239–242.  Prado, J. L.; Alberdi, M. T.; Azanza, B.; Sánchez, B.; Frassinetti, D. (2001), "The Pleistocene Gomphotheres (Proboscidea) from South America: diversity, habitats and feeding ecology", in Cavarretta, G.; Gioia, P.; Mussi, M. et al., The World of Elephants - Proceedings of the 1st International Congress, Rome October 16-20 2001 (Rome: Consiglio Nazionale delle Ricerche), 337–340, , PDF link, retrieved on 25 July 2008.