It used to be common for brits to americanize thier voices for US consumption. Patrick Stewart in ST:TNG also toned down his own (very english) english for the US. Although it must be pointed out that Marina Sirtis own voice is wildly cockney and as such should not be treated as English any more than scouse.

As far as annoying goes, if you though Costner was bad, the award for the most annoying has to go to Dick Van Dyke in Mary Poppins
Or for a more modern slant, Keanu Reeves in Much Ado about Nothing.
Personally, the bad accents are more funny than annoying.
For a good accent job, check out Rene Zellwegger in Bridget Jones Diary.

The thing I find most annoying is the way Hollywood will rewrite history as pass it off as accurate. Take U571, Braveheart, etc etc even Pocohontas for goodness sake!
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BTW - Pocohontas died in Endland after being there for seven whole months, not very Disney friendly. http://www.linkline.com/personal/xymox/roh/poca.htm