From Bill Haun: Interestingly, there is a place here in the U S---Smith's Island where the native people speak in Elizabethian dialect still.   It is off the coast of Maryland in the Chesapeake Bay area. Written up in National Geographic many years ago.

It's also said that they speak with Shakespeare's accent on Tangier Island, off the coast of Virginia (also in the Chesapeake bay). The reason is that the settlers to these islands did indeed come from Shakespeare's neighborhood at the time that Shakespeare was alive (we're celebrating the 400th anniversary of the founding of Jamestown in 2007; Shakespeare was near the peak of his craft in 1607), and, being islands, the accents and turns of phrase were preserved (though mass television radio broadcasts are now diluting it).

So, I guess you really could say that Shakespeare spoke with an "American accent" (if you visit the right niches of America).