That was the question recently posed by the Telegraph.
From the piece...
Inspired by working with Kevin Spacey, Sir Trevor Nunn has claimed that American accents are "closer" than contemporary English to the accents of those used in the Bard's day.
The eminent Shakespearean scholar John Barton has suggested that Shakespeare's accent would have sounded to modern ears like a cross between a contemporary Irish, Yorkshire and West Country accent.
Others say that the speech of Elizabethans was much quicker than it is in modern day Shakespeare productions.
Well, now you can judge for yourself.