Several decades before DNA testing of Native Americans began, LDS scholars were recognizing that the popular hemispheric view of the Book of Mormon did not square with the text, and that it allows for or hints at the presence of other population groups not descended from Lehi or the Mulekites. You can lay this information out in plain detail for the critics, but they go on as if the Book of Mormon has been proven false by showing that typical Native Americans appear to be predominately descended from Asians (never mind that Jaredites may have been Asian and may be expected to have exerted a much larger impact on the New World gene pool than Lehi's group). I think they are so excited about the idea of "finally" destroying the Book of Mormon and LDS faith that they just can't see straight on this issue. Or perhaps they are seeing too straight - straight down a microscopic tunnel focused on a few strands of Asian DNA.
Yes, the finding of Native American DNA haplotypes more related to Asian peoples than modern Jews greatly reduces the likelihood that Native Americans only have ancient Jews as their ancestors - but that's not what the Book of Mormon requires at all. It does not rule out the possibility that an ancient Jew named Lehi entered an inhabited continent 2600 years and founded a colony that grew and became locally but not hemispherically significant. It's critical to understand the actual scope of the Book of Mormon and its relationship to actual LDS doctrine. On this topic, here is an excellent excerpt from John Sorenson and Matthew Roper in their paper, "Before DNA":
How does this [limited] geographical picture square with traditions held among the Latter-day Saints about the scenes and peoples involved in Book of Mormon events?The scope of Book of Mormon geography has long been an issue open to debate and discussion, rather than something settled by official Church doctrine. Careful reading of the text has led to the clear conclusion that the text deals with a limited geographical scope, and it appears to best fit within a small part of the hemisphere in Mesoamerica. LDS scholars for several decades have been recognizing this - it's not some new retreat to escape the challenge of DNA evidence. The anti-Mormons don't want it to be so. They want investigators and weak members to think they have decimated the Book of Mormon and the foundations of the Church by showing that there is heavy evidence of Asiatic origins among Native Americans. They are victims, I'm afraid, of microscopic tunnel vision.
We face a lack of detail in our historical sources as to what the earliest Latter-day Saints thought about Book of Mormon geography. Even so, there is little question that generally an obvious interpretation was in many readers' minds. The "land southward" they considered to be South America, the Isthmus of Panama was "the narrow neck," and North America was thought to be the "land northward." However, there is no evidence that in the early years any detailed thought was given to geography. Actually, the Book of Mormon was little referred to or used among church members in the first decades except as a confirming witness of the Bible. The writings or preaching of some of the best-informed church leaders of that day show that they did not read the text carefully on matters other than doctrine. For instance, no statement shows that anyone read the scripture closely enough to grasp the fact that the plates Mormon gave to Moroni were never buried in the hill of the final Nephite battle.
In 1842 a best-selling book by explorer John Lloyd Stephens was read by Joseph Smith and associates in Nauvoo [Incidents of Travel in Central America, Chiapas, and Yucatan (New York: Harper & Brothers, 1841)]. Their reading prompted an extensive review of the book in the Nauvoo newspaper, the Times and Seasons. (No author is listed, but Joseph Smith was editor in chief with John Taylor as managing editor.) Stephens's was the first book in English reporting great ruins in Central America. It strongly impressed the newspaper writer (whoever he was), for on 15 September the paper reported, "We have to state about the Nephites that . . . they lived about the narrow neck of land, which now embraces Central America, with all the cities that can be found." ["Extract," Times and Seasons, 15 Sept. 1842, 914.] Stephens's new information obviously was causing the leadership in Nauvoo to think of Nephite geography in a new way. Two weeks later they continued to exult in their study of what was for them "the latest research": "We have [just] found another important fact relating to the truth of the Book of Mormon. . . . The city of Zarahemla . . . stood upon this land," that is, Central America or Guatemala, which "once embraced several hundred miles of territory from north to south." ["Zarahemla," Times and Seasons, 1 Oct. 1842, 927.] Since Zarahemla was located in the land southward, their new insight put the land southward to the north of Panama. The new thinking inferred that South America was of little or no significance for Book of Mormon geography. The further inference is that the new thinking was that an area much smaller than the entire hemisphere could satisfactorily serve as the scene of the chief events in the Nephite record.
In the long run, nevertheless, the Stephens-stimulated view of Central America as the Book or Mormon heartland did not prevail among the Saints generally. The new implications were apparently overwhelmed by the inertia of the old belief in a whole-hemisphere geography. Orson Pratt, who was separated from the church during 1842 when the new thought on this topic was stirring, seems to have continued to believe in the original geographical theory. His views along those lines are reflected in the geographical footnotes that he added to the 1879 edition of the Book of Mormon. His opinions led several generations of readers of the scripture to assume with him that only the Nephites and Lamanites of Mormon's account occupied the Americas, from the Arctic to the Antarctic, at least during Book of Mormon times. By the beginning of the 20th century, likely not more than a handful of readers of Mormon's book questioned the interpretation that Lehi landed in Chile, that Panama was the narrow neck, and that the final battle of the Nephites took place in New York.
Anecdotal evidence (there are no systematic data) suggests that even now, after church members have been reading the Book of Mormon for a century and three-quarters, a large number of readers continue to assume the whole-hemisphere view of Book of Mormon geography. Moreover, some unbelievers insist in their anti-Book of Mormon propaganda that this view was and is completely orthodox (which makes their criticisms more damaging). But the proportion of Saints who still accept that antiquated geography is irrelevant in light of the decisive information in the Book of Mormon. The text itself gives an unmistakable picture of a very restricted territory. And as President Joseph Fielding Smith said, "My words, and the teachings of any other member of the Church, high or low, if they do not square with the revelations, we need not accept them." [several footnotes omitted]
For more info on the DNA and Book of Mormon issue, you can read a variety of papers on DNA at FARMS (links at the bottom of the page). I've also got a rather lengthy essay on the topic of DNA and the Book of Mormon, where, among other things, I give more detail about LDS teachings and internal evidences in the Book of Mormon for other peoples in the Americas.
The author of one typical anti-Mormon paper states, "The facts are a comparison of the DNA of the Middle East has been made with the DNA of Native Americans [and] there is not a match." This is supposed to devastate the Book of Mormon, and that's how the anti-Mormons present the case. It's unscientific. Why should we expect a match? Even if modern Jews had the same distribution of mtDNA and Y chromosome haplotypes as ancient Hebrews in Lehi's day, and even if Lehi and all the others he brought with them fit squarely within that distribution, why should we expect a genetic "match" between modern Native Americans and modern Jews? Such talk relies on errant assumptions about the scope of the Book of Mormon and the requirements of the text. The author of that particular paper has seen the evidence that the Book of Mormon deals with a limited geography, but insists on clinging to the hemispheric assumption - it's tunnel vision.