As an American actor, I have never had a problem hearing Shakespeare's words delivered in high stage, standard American English. Recently, I've heard that the American accent is closer to how Elizabethan Englishmen spoke than modern British English is. But, the main argument I would make in favor of keeping the accent American for an American audience (and an American cast) is that it removes one possible obstacle between the meaning of the play and its audience. The audience knows this is an English play from an earlier time and adjusts accordingly to the language. Why not keep the accent familiar?
Having said that, I will admit that we Americans are watching a LOT of British tv, and the accent there doesn't seem to be a problem. Then again, the language of Downton Abbey is pretty modern and familiar; even the breath-takingly great Maggie Smith does not speak in verse or iambic pentameter. Other things I will admit, though I question their relevance: Mark Rylance was born in New Jersey, and Damian Lewis is just as convincing as an American as he is as an Englishman. But this next bit IS relevant.
Richard Burton, whom people differ on but who was more astute, I think, than people give him credit for, Richard Burton, I say, said in a radio interview I heard years ago that one of the best Hamlets he had ever seen/heard was in San Francisco, and it was Robert Preston. Now, Robert Preston was wonderful, but how many people think of him as a Shakespearean? Apparently, he was. I believe just about any good actor can play Shakespeare if he/she avoids descending into Shakespeare Panic, and plays AT Shakespeare instead of just playing the character to hand. And the Great One, Jackie Gleason, did you know he longed to play Hamlet, but no one would go along with this longing? Today, being such a big star, he could have done it easily, no problem. I would have LOVED to see Gleason's Hamlet. Do you remember The Poor Fool? This guy had range.
John Goodman: I don't know what he aspires to, but whatever it is, I hope he gets the chance. His talent is just dazzling. But, and forgive the tangle of this post, as I'm trying to re-create one that I lost, which had an impeccably logical thread, and this one not so much, but Richard Burton's Hamlet needs to be discussed. Not now, but at some point. I remember hearing Burton recite "To Be or Not To Be" in German. Now, there was something to chew on. The language was so firm and rich and assertive, Hamlet sounded like a general giving himself marching orders, instead of some dreamy, vacillating wimp.