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This book is a popular version of one of Ehrman's earlier, more scholarly books, Orthodox Corruption of Scripture. Both content and method have transferred over substantially, particularly Ehrman's major weaknesses, which are:
1) a lack of familiarity with broader defining contexts (eg, Jewish Wisdom theology, which, for example, solves the alleged "problem" he sees in Heb. 1:3 and resolves it in favor of the "manifests" reading -- 56) that would weight down heavily in terms of solving alleged problems he "discovers";
2) treating problems in the text as though they are far more serious than they really are.
The bulk of this book is unobjectionable history without any scent of controversy. As such it has value as "Textual Criticism 101" but it has numerous quite serious problems interspersed with the neutral narrative. I refer readers to a review here and will use some of the points made there by Daniel Wallace as a basis for my own analysis, which will NOT include commentary on such matters as we have covered elsewhere in depth (eg, Q and Marcan priority).
Wallace noted that the book’s very title is a bit too provocative and misleading though: Almost none of the variants that Ehrman discusses involve sayings by Jesus! One may suspect that editors, not Ehrman, chose this title for salability.
Ehrman, as noted, has a tendency to simply create problems where none exist, and then expects readers to share his overzealous worry. Semantics dictates that his concern to have the "very words"  of the original, inspired text is misplaced. Communication is simply not that difficult to achieve, and ancient people simply were not that concerned with precise verbiage. Nor does it stand well as a claim made in a book where he claims to be solving and explaining the very things he says are problems.
Furthermore, Ehrman's comments find no parallel in the works of secular textual critics, who I have yet to see say things like, "we don't have the originals"  or "can we be sure that all the copies were all 100 percent correct"  of something like Tacitus' Annals, and then make some issue over it as though it was a problem for knowing what the Annals said.
Wallace's conclusion is well worth quoting in full at this point:
In sum, Ehrman’s latest book does not disappoint on the provocative scale. But it comes up short on genuine substance about his primary contention. Scholars bear a sacred duty not to alarm lay readers on issues that they have little understanding of. Unfortunately, the average layperson will leave this book with far greater doubts about the wording and teachings of the NT than any textual critic would ever entertain. A good teacher doesn’t hold back on telling his students what’s what, but he also knows how to package the material so they don’t let emotion get in the way of reason. A good teacher does not create Chicken Littles.
One is rather suspicious of how Ehrman cites Celsus for the point that Christianity's church was "largely made up of the lower, uneducated classes."  Aside from the needed corrective of Meeks and Judge that Christianity early on was "top heavy" (relatively speaking) in the educated middle class, it remains that 99% of people who lived in the days of Celsus were of the "lower, uneducated classes" and so this sort of thing was merely a matter of demographic necessity for a movement to exist and not any sort of indication that the movement itself was rooted in stupidity. Celsus was not being honest, and Ehrman in following him is also not being honest, or more likely, does not have the needed knowledge beyond his specialty area of textual criticism to see his error.
An example of how Ehrman fails to grasp context as a problem solver is illustrated in his exercise on page 48 in which he presents the reader with the run-on word, "lastnightatdinnerisawabundanceonthetable". He asks rhetorically if this was "a normal or a supernatural event" and pretends that problems like these exist in the Biblical text (but gives no examples from there of an actual problem because of this).
Ehrman does not even consider that we can figure out which reading is right by checking context. If the sentence is by itself, there's no reason for us to check any further either way. But if it is followed by, "unclehenrytriedtostabitwithhisforkbutthenitdidthewatusi", or by "therewerelotsofmeatsandcheesesandbreads", then we know what to read the run-on as either way.
Ehrman discusses a few specifics of textual issues, some of which we have covered before: John 8, for example. Wallace's review notes others and we agree with the conclusion, which we can put no more eloquently:
...the idea that the variants in the NT manuscripts alter the theology of the NT is overstated at best. Unfortunately, as careful a scholar as Ehrman is, his treatment of major theological changes in the text of the NT tends to fall under one of two criticisms: Either his textual decisions are wrong, or his interpretation is wrong. These criticisms were made of his earlier work, Orthodox Corruption of Scripture, which Misquoting Jesus has drawn from extensively. Yet, the conclusions that he put forth there are still stated here without recognition of some of the severe criticisms of his work the first go-around. For a book geared toward a lay audience, one would think that he would want to have his discussion nuanced a bit more, especially with all the theological weight that he says is on the line. One almost gets the impression that he is encouraging the Chicken Littles in the Christian community to panic at data that they are simply not prepared to wrestle with. Time and time again in the book, highly charged statements are put forth that the untrained person simply cannot sift through. And that approach resembles more an alarmist mentality than what a mature, master teacher is able to offer. Regarding the evidence, suffice it to say that significant textual variants that alter core doctrines of the NT have not yet been produced.
It is particularly sad that Ehrman revives Elizabeth Cady Stanton and emphasizes the large number of variants  without concern for laying out their importance or meaning. It is also unfortunate that Ehrman uses standard objections about texts like 1 Tim. 2:11-15 and 1 Cor. 14 [184f] on women (see here, and Matt. 27:24-25 on the reaction of the Jews (see relevant part here). This is yet another example of how Ehrman's lack of knowledge outside his specialty leads to false conclusions.
One new example is how Ehrman thinks a change from "the Jews" to "Judaea"  somehow reflects anti-Semitism. Socially, in that day, there is no difference whatsoever, since "the Jews" is "the Judeans" -- it is a national and not a religious term. Ehrman is reading modern understandings of identity into the text.
In conclusion, we find this book in some ways to be more of an ideological treatise that takes too many liberties when it comes to offering half-truths and in finding significance where none exists.