Mormons believe that a group of a couple of dozen Israelites immigrated to the American continent in about 600 B.C. and that they are among the ancestors of some of America's indigenous tribes. For starters, this small group of people from Israel were not part of a "lost tribe." Secondly, within a relatively short time, they intermarried with the many people who already occupied the American continent when they arrived, so you would definitely not expect to see any pure Hebrew blood lines among the American Indians of today. It's possible that Sitting Bull's hundredth great-grandfather was of Jewish descent, but to assume that he belonged to some Jewish lost tribe is really very illogical.
The question to be argued really isn't, "Are today's Native Americans of Middle-eastern ancestry?", but "Is it possible that a small family from the Middle-east could have settled on the already populated American continent 2600 years ago and left no genetic evidence of their existence?" Genetic drift alone would explain how Lehi's (the central figure at the beginning of the Book of Mormon story) haplogroup would almost certainly have disappeared after just a few generations. If Lehi and his family had arrived on an empty continent, it would be a different matter entirely, but we know that wasn't the case.
Lineages simply disappear over time. There can be a tremendous discrepancy between the DNA of people who lived 300 years ago and the people who live now. One very good example of this is found in the "deCODE Project" in Iceland. Recent research shows that the vast majority of today's Icelanders are descended from a tiny percentage of people who lived less than 300 years ago. Many people living there as recently as the mid-1700s have no genetic lineages represented in Iceland's population today.