One of the best examples I've found that starkly demonstrates it, is the Kenneth Branagh production of Hamlet. The Americans whom I remember from that cast were Billy Crystal as The First Gravedigger, Robin Williams as Osric, Jack Lemmon as Marcellus, and Charlton Heston as the Player King. There was a stark contrast between them and the British actors. The British actors "recited" their lines; the Americans spoke their lines. The only time that Mr. Branagh spoke his lines was when he had a scene with an American actor. Charlton Heston seemed the least skilled among the Americans, but otherwise, it seemed as if there were two separate casts in the film. The other exceptions that I recall in that cast were Gerard Depardieu as Reynaldo, and Julie Christie as Gertrude. Julie Christie said that she had never before performed Shakespeare. Apparently, she approached the material as she would any prose, and Mr. Depardieu, for the several lines he had, well, I'm sure you can imagine how he played it even if you haven't seen the film.
I recently read Kirk Douglas's autobiography, The Ragman's Son. He mentioned the American actor, Edmund O'Brien in the film of Shakespeare's Julius Caesar in which Marlon Brando played Mark Antony. Mr. Douglas mistakenly said that Mr. O'Brien played Cassius (played by John Gielgud; O'Brien played Casca). Nevertheless, he further said, "Although underrated, Edmund O'Brien was one of the best Shakespearean actors. In simple-sounding American speech, Edmund made the words of Shakespeare sound as if they had just occurred to him, really came out of him."
...the words as if they had just occurred to him - yes. I'm willing to claim that it was the contribution of the members of the Group Theatre who greatly influenced the American acting tradition; Laurette Taylor notwithstanding.