If you want to attack the linguistics side of this question a bit more, THIS story based on THIS research is one of the best sources out there. Of course, where and how the colonial footprint ties in with the linguistics side of the research is not clearly addressed. Either way, I think it may be more of a question for /r/linguistics.

I doubt the basic idea here is true, or at best it is hugely simplified. There's some interesting stuff about how the colonial "cores" formed, before 1750 or so, in D.W. Meinig's book The Shaping of America, Volume 1: Atlantic America, 1492-1800. He identifies about 20 core colonial societies along the eastern seaboard around 1750 (mainly those that were or became part of British America) and points out how all of them drew from various source regions in Europe and Africa, mostly in a complex way.

He points out how it isn't enough to identify early cores of American settlers with specific source regions in Europe. At least as important was the process by which people migrated to America. A large proportion of the people who migrated to America had already moved from their native regions in Europe. Many had left rural areas and moved to cities before migrating to America. Many had lived in various regions and various countries in Europe before moving to America. As a result it isn't enough to know the place of birth, port of depature, and colony arrived at if you want to understand how the colonial cores acquired the characteristics they did—culture, ethnicity, dialect, accent, religion, etc.

Consider a few examples (drawn from Meinig): A laborer from the English Midlands who spent ten years in London before moving to America may be more an Englishman "in a larger sense" than his Midland origins might suggest. A business agent from Glasgow who migrated to Maryland only after a long tour of duty in the West Indies may be as much a "British American" as a Glaswegian. A man who left a Tipperany cottage when young for the docks of Kinsale may no longer be an Irish peasant. And so on—Huguenots often spent years moving around the Channel ports of England, Palatine Germans frequently got "marooned" in London for months or years. The whole northwest European "cultural hearth" was in a kind of flux, and often turbulent—at least, or especially for people who ended up migrating to America.

And then there was the actual crossing of the Atlantic. Most emigrants came alone, or with a few family members. Very few came in larger cohesive groups. And then in America there was often more time spent in ports and migration within and between regions.

In short, people naturally brought their accents, dialects, languages, customs, etc, with them to America, but most of the time these did not long survive in America because the person would be thrown together with other people with different backgrounds. Accents "converged" in complex ways that usually cannot be linked to specific source regions in Europe, at least not in a simple way. Patterns emerged from the chaos and colonial cores formed their own particular identities, but many elements played into how this happened.

All that said, some colonial cores were more homogeneous than others, epsecially (French) Canada and New England. But even in these places people came from many different source regions in Europe, and under strong selective pressures that favored only certain kinds of people (eg, Catholics, Puritans). Most colonial core societies were far more pluralistic in origin and development. South Carolina, for example, in which there was a coastal zone with a strong English-Barbados background as well as relatively large numbers of Huguenots and Scottish Lowlanders. Then in the backcountry there was a complex mixture of English, Ulster Scot, German, and others. And slaves whose backgrounds were from diverse regions of Africa plus long periods in the West Indies. And there were Native American influences as well.

The African aspect to all this is often overlooked. But in most of the colonial cores Africans were a significant part of the population from the start. Even when there was harsh segregation between Africans and Europeans there was also close contact, especially in the early period of a colonial region, when its defining qualities, like accents, took root, influencing later migrants. In addition, during the early phase of some colonies, like South Carolina, there were large numbers of Native American slaves, here again in close contact with the European population.

All this is to say I am doubtful that you could map colonial areas of the eastern US to particular areas in Europe in a meaningful way—at least beyond very broad brushstrokes (like say French Canada drew more from France and Catholicism). Or put another way: it may be possible in some cases to map American colonies to source regions in Europe, but whether one could then say the colonial accent was more heavily influenced by the European source region(s) is questionable at best, I think.

The book I've been talking about is The Shaping of America, Volume 1: Atlantic America, 1492-1800, D.W. Meinig. Sorry for the length of this post, I got carried away!

David Hackett Fischer wrote an excellent book, Albion's Seed, which traces different early American regional cultures to different waves of immigration from the British Isles in the 17th and 18th centuries. I highly recommend it.

That said, as /u/Rittermeister noted below, the narrative that particular regions of North America were settled mainly by people from particular regions of Europe is rather over-simplified and the demographic reality by the early 19th century is rather more complex.

There have been efforts to map American dialects over the last century or so: http://aschmann.net/AmEng/ is a good place to start.

I read that the south got more immigrants from southwest England and the north got more from east anglica.