In a thread like this I’m always afraid that I’ve somehow missed the point, so forgive me if I’m getting this wrong, but…
Naveen Andrews (a Brit) on Lost playing an Iraqi guy who speaks English.
Whoa - should I have used a smiley? I was totally joking. (How’s that for a US adverb)?
Judy Davis has played English many times, though I believe she’s from New Zealand.
Wendy Hiller was Russian in Murder on the Orient Express. Laurence Olivier was Polish (or some other indeterminate Easter European type) in The Boys from Brazil and I think he played German in Marathon Man, although I haven’t seen it.
Apparently every U.K. actor with even slightly non-pasty-white skin has played Indian or Middle Eastern roles complete with accents – Ben Kingsley, John Rhys Davis, Alfred Molina, and Alexander Siddig, for a start.
Conversely, the actors with pasty-white skin get to play Teutonic. Jeremy Kemp owns the Nazi subcategory all by himself. Also Alan Rickman and Jeremy Irons as the Gruber brothers had German accents, I believe; at least, I know Rickman did.
All the Monty Python folks have done Australian accents, most notably in the “Bruces” sketch.
And of course, for years Patrick Stewart astounded us with his awesome French accent playing Jean-Luc Picard. 
To be fair, Kingsley’s father was of Gujarati descent and was an Ismaili Muslim, though how he ended up with the first name of Krishna (originally named Krishna Bhanji) is beyond me. Siddig (full name Siddig El Tahir El Fadil El Siddig El Abderahman El Mohammed Ahmed El Abdel Karim El Mahdi) is half Sudanese. Alfredo Molina is Spanish and Italian, so he came by his Mediteranean looks naturally. Rhys-Davies is, well, Welsh on both sides, but he spent some of his youth in Africa, so he STILL has more claim to playing non-European roles than did, oh, Marlon Brando (nee “Marlon Brando, Jr” and a typical US Western-European mutt) playing a Japanese gent in “Teahouse of the August Moon.” Or Iron Eyes Cody (born Espera de Corti and all Sicilian) playing a Native American.
(all factoids courtesy Wikipedia, the most fabulous trove of useless and possibly incorrect information since *The Book of Lists * and The People’s Almanac.)
I came in here to mention Alan Rickman. Someone in another thread about accents not too long ago mentioned the scene in Die Hard where his character faked an American accent. So we have a British actor speaking with a German accent pretending to be an American!
Two of the people you list are actually qualified to play the roles for more than just their skin tone.
Ben Kingsley, real name Krishna Bhanji, father Muslim Indian and mother English.
Alexander Siddig, real name Siddig El Tahir El Fadil El Siddig El Abderahman El Mohammed Ahmed El Abdel Karim El Mahdi and was born in the Sudan to a Sudanese father and English mother and hisuncle was Sudanese Primeminister. He’s also a practicing muslim.
You have a point with Rhys-Davies, even though he was raised in Africa, he’s welsh through and through, and Alfred Molina who whilst Spanish-Italian by descent isn’t nearly as qualified as Krishna or Siddig.
Australian says wiki… but you know… we do actually speak English over here (and I have it on some authority that even some of the Australians speak it too). 
Smart-assing aside, not everyone in NZ says “fush and chups” nor Aussies “feesh and cheeps”… my speech / accent has been mistaken for English a number of times both in NZ and abroad, and I’m Kiwi through and through – I just speak proper.
If I could train myself to speak in accent other than my own, I’d like either RP or Kiwi… Stereotypical Kiwi sounds cool.
Mmmm… 
Two Youtube clips with good stereo-typical Kiwi accents:
Hmm… Maybe I’ll have to re-evaluate.
Lynn of Tawa! Argh! Now there’s something I never wanted to be reminded of.
WHAT?!? Oh for Christ’s sake… :smack:
I always think of John Hillerman, an American actor whom I was first exposed to on Magnum PI, where he had a passable British accent. Imagine my surprise when he faked such a great American accent in Bazing Saddles! 
A few off the top of my head:
Paul Freeman (British) plays French in Raiders of the Lost Ark; in fact, he seems to make a something of a specialty of playing non-British Europeans
Ben Kingsley (already mentioned), Ralph Fiennes (British) and Liam Neeson (Irish) all play German in Schindler’s List
Liv Ullman (Norwegian, born in Tokyo) plays Dutch in A Bridge Too Far
Paul Scofield (British) plays German in The Train
Jack Hawkins (British) plays, er, Danish(?) in Zulu
Half the cast of Casablanca seems to be playing nationalities other than what they were. Claude Rains (British) plays French; Sydney Greenstreet (British) and Peter Lorre (Austria-Hungary, now Slovakia) play Italian, and I’m probably forgetting a few others.
Dr. Zhivago, being set in Russia and all, has a raft of non-US actors playing Russians: Omar Sharif (Egyptian, listed as ‘Franco-Egyptian’ on IMDB), Julie Christie, Alec Guinness, and Tom Courtenay (all British), among others.
There’s a thorough discussion of the actors in Casablanca and what their real nationalities are versus what nationality they play in Wikipedia:
Only three credited actors in the film are American.
Yep, I know all that, of course! But the thread’s about accents, and all of the above speak with British accents, and have been cast in a plethora of non-UK-accents that aren’t just their own (or relatives’) backgrounds. Their appearance and talent allow them this flexibility. Siddig may be part Sudanese, but he’s played Syrian, Algerian, Carthaginian, and, erm, Heavenese (as the Angel Gabriel :)). Molina’s roles have included Iranian, Russian (both ‘regular’ and Yiddish accent flavors), Mexican, Belgian, Egyptian. And Ben Kingsley’s played Indian, Polish, Austrian, Italian, Egyptian, and Russian! Pretty amazing, really.
(Of course, whether they play them all believably is a different story – I’m not qualified to judge how accurate their accents are.)
Actually, I don’t think so. I think his accent is used as a deliberate reminder that he’s not an ethnic Russian but a Balt.
Well, in that case how about Sean Connery’s “Egyptian” accent in the original *Highlander *movie?
Oh, that was brilliant.