Living as I did recently very close to Appalachia, and having had a professor a bit odd in his adopted obsession for it, I thought I'd draw attention to this post over at Language Log.
Every linguistic change is brought about by the majority of a language's speakers regardless of their knowledge of its history. This may sound like a truism, but aside the last five hundred years of changes, our language is exactly as it was in 1505. (Which isn't nearly as drastic a set of changes as those of the previous 500 years.)
Anyway, hence the gleaming truth of Sally Thomason's observation: "There are said to be features of Shakespeare's English that are preserved in Appalachian English but not in Standard English; but they would be noticeable only because they have vanished from Standard English. The many features of Shakespeare's English that remain in Standard English are not noticeable: they're just ordinary." Yet they're there. They're just not interesting enough to pass for urban legend. It'd be like saying, "Did you know that the language of Shakespeare's descendants has only one word for 'snow'??"
The error is not is the observation of many "archaisms" of Appalachian ("Southern Mountain") dialects. It is in forgetting that there are also many in standard modern English. For example, take the sentence: "Alfred was grim late in his life; his king dead, he sang of frost and blindness and death." This sentence is 100% archaic. With very few differences in pronunciation, it is exactly the same as it would have been a millennium ago. It is in fact an "Old English" sentence. (See more cool sentences like this, and hear their pronunciation, here.)
See also this LSA article, which calls the association of Appalachian and Shakespearean English one of "the most persistent myths about language."
For those interested (my old professor, for instance), here is an annotated bibliography of Appalachian English.