I grew up in Appalachia, as did my parents and relatives. It is a culture I know well, and five of my inspirational novels are set there. Going to college, teaching, and giving many presentations over the years, I've been very aware of my Appalachian speech patterns and been discriminated against because of them. Perhaps, in some small way, this played into why I became a writer. I might not have a general American accent when I speak, but on paper, it's not noticeable.
There is some truth in the fact that Appalachian speech has retained more of the ancient British vocabulary than other places. For years, people who lived there were isolated, and the rugged, mountainous terrain helped with this. However, to say that it is like Chaucer's or Elizabethan English would be a great exaggeration. Language is fluid and always changing. Still, there're plenty of words in common use in the Southern Appalachians that can be traced back to Shakespeare's day or before. For example, a university English course I took on some early English plays had the word "flyflapper" in one. None of my classmates knew what it meant, but my mother had always used it. It's a fly swatter.
Some other such words are britches (pants), poke (bag), sallet (greens like kale or mustard), afeared (afraid), backward (shy or slow to learn), counterpin (bedspread, counterpane), directly (soon, in a little while), falling out (disagreement, grudge), far-fetched (unreasonable), frolic (party, celebration), hanker (yearn, desire), heyday (a period of popularity), hobble (walk with a limp), ignoramus (an ignorant, uneducated person), lambasted (attacked, often verbally), liable (apt to, likely), looking glass (mirror), nigh (close; nearly), piddle (dawdle, do something at a leisurely pace), pilfer (underhandedly steal), plunder (ramble through goods), pone (swollen knot on the body; cake of bread), sheepish (meek, timid), sot (a drunk), tickled (pleased), and many more. Of course, there are differences in pronunciation, too, but that's a whole other article.