I videotaped the show off the air way back when. The tape’s crappy by today’s standards but it’s still one I enjoy hauling out to watch now and then.
You’re probably thinking they had English accents, whereas a lot of southerners would have actually come from Scotland or the north of Ireland. If you listen to a modern Donegal accent in particular it’s fairly easy to hear a connection to the southern US.
Also, bobo t, you may be thinking that it was just the American accents that changed and not the British. (We get this question on the SDMB every once in a while.) All dialects change all the time. They change slowly perhaps, but they inevitably change. The American (and Canadian and Australian and New Zealander) accents of today are just as much different from the British accents of 400 years ago as the British accents of today are from the British accents of 400 years ago. Staying in one’s homeland doesn’t mean that a population’s language remains unchanging. For any particular difference between American and British dialects today, it would take some research to discover whether the British dialects of 400 years ago are closer to the present American dialect or the present British dialect.
Wow, I’d forgotten even starting this thread! Thanks for all the replies.
And yes, it’s entirely possible my teacher was pulling our legs. He “did” a few of us, and then moved onto something else, so maybe he just chose those of us who had something distinctive in our dialects but would have gotten the others wrong. Or maybe he cheated and memorized our student files and was a good cold reader. 
As a side question: what’s the deal with the woman who plays Charlotte on Lost? I could have sworn that she was using an Australian accent, and I was shocked recently when another character referred to her as British. But looking up her bio, I see the actress is from Coldham’s Lane, England. I’ve *never *mixed up an Aussie for a Brit before! I think. I’m shaken. Is that a real Coldham’s Lane dialect, or is she using a different voice for her character?
Coldham’s Lane, Cambridge - quite why they’ve listed the entire address, I don’t know! Listening to her here, it’s a big jumble. There’s East Anglian roots still audible (“Oi did” at 4:11 is a great example, and plenty of dropped consonants and glottal stops), but mainly overridden by the generic middle-class ‘English Accent’ so loved by Americans, and with various tinges around the edges of American influences, as one might expect of someone who’s lived and worked there for years.
Yeah when Fearghal Sharkey or Van Morrison sing they seem to bridge the gap with ease, more so than a singer from the south would.
Modern sociolinguistic thought tends to believe (and this partially answers BrainGlutton’s question about why AAVE [“Ebonics”] and standard English exist side by side*) that dialects and accents are maintained as a marker of identity. Speaking with a certain accent says “I am a part of X group”. Sociologically speaking, the act of a Georgian speaking with a Deep South drawl is similar to that of a South Indian person eating idlis or an anarcho-punk wearing a black bandana.
*however, it is necessary to note that AAVE is not just an accent-- it has a grammar separate from standard English.