FairMormon is a non-profit organization dedicated to providing well-documented answers to criticisms of the doctrine, practice, and history of The Church of Jesus Christ of Latter-day Saints.
Questions: What are the criticisms related to the Book of Mormon and DNA?
DNA attacks against the Book of Mormon account fail on numerous grounds
Few criticisms of the Church have received as much media attention as this criticism, with so little thought and science being applied to the question. DNA attacks against the Book of Mormon account fail on numerous grounds.
It is important to realize that critics of the Book of Mormon base their arguments on DNA data that has never been shown to be even relevant to the issue of Book of Mormon genetics, let alone conclusive. Such critics have cobbled together DNA data gathered from unrelated studies to produce arguments with the appearance of scientific weight but having no real significance. No genetic studies have been designed and performed to test the hypothesis that Native Americans were of Lehite descent and that this inheritance is detectable today.
DNA samples taken from modern Native Americans do not match the DNA of modern inhabitants of the Middle East. Critics argue that this means the Book of Mormon's claim that Native Americans are descended from Lehi must be false, and therefore the Book of Mormon is not an ancient record as Joseph Smith claimed. 
"Recent attacks on the veracity of the Book of Mormon based on DNA evidence are ill considered." Various geographical models introduce issues unique to each model, but the DNA data is no where as conclusive as some claim, regardless of the geographical model chosen.
Critics tend to opt for the most naive, ill-informed reading possible of the Book of Mormon text, and then cry foul when the Saints point out that they have given much thought to these issues and come to more nuanced conclusions that are more faithful to the Book of Mormon text than the critics' poorly-considered caricatures.
Critics do not provide the "whole story" of the DNA data, and seem to want to use the certainty which DNA provides in modern crime-solving as a springboard to trick the Saints, the media, and investigators into thinking that their historical DNA conclusions are as solid.
DNA data tells us nothing which we did not already know from archaeological data
In fact, DNA data tells us nothing which we did not already know from archaeological data—at present, the human settlement of the Americas is thought to date thousands of years before the advent of Lehi. Many of these settlers have links to east Asia. None of this is news, and none of it threatens the Book of Mormon's status as authentic history.
But, the critics hope that their listeners will be awed by the banner of DNA science, and conclude that something more impressive is going on. Informed members of the Church have not been persuaded by their tactics, and much has been written to help non-specialists understand the "numerous and complex" issues in the fascinating and valuable science of genetics.
- Criticisms regarding DNA and the Book of Mormon have been raised in the following publications: Richard Abanes, Becoming Gods: A Closer Look at 21st-Century Mormonism (Harvest House Publishers: 2005). 73 367 n.131-135. ( Index of claims ); John Dehlin, "Why People Leave the LDS Church," (2008).; Hank Hanegraaff, The Mormon Mirage: Seeing Through the Illusion of Mainstream Mormonism (Charlotte, NC: Christian Research Institute, 2008), ?.; Walter Martin, The Kingdom of the Cults (Revised) (Minneapolis: Bethany House Publishers, 1997), 202. ( Index of claims ); John Dehlin,"348-349: Simon Southerton, DNA, Lamanites and the Book of Mormon," Mormon Stories Podcast (21 May 2012).
http://mormonstories.org/348-349-simon-southerton-dna-lamanites-and-the-book-of-mormon/; MormonThink.com website (as of 4 May 2012). Page: http://mormonthink.com/book-of-mormon-problems.htm; Thomas W. Murphy, "Lamanite Genesis, Genealogy, and Genetics," in American Apocrypha, ed. Dan Vogel and Brent Lee Metcalfe (Salt Lake City: Signature Books, 2002), .; Thomas W. Murphy and Simon G. Southerton, "Genetic Research a 'Galileo Event' for Mormons," Anthropology News 44/2 (February 2003): 20; Simon Southerton, Losing a Lost Tribe: Native Americans, DNA, and the Mormon Church (Salt Lake City, UT: Signature Books, 2004) 1–207. ( Index of claims ); Tower to Truth Ministries, "50 Questions to Ask Mormons," towertotruth.net (accessed 15 November 2007). 50 Answers
- John M. Butler, "Addressing Questions surrounding the Book of Mormon and DNA Research," FARMS Review 18/1 (2006): 101–108. off-site wiki