Obviously inspired by this opposite thread.
I think this one is going to be more difficult, because we for the most part already know how most actors sound, so any accent they do has to be 100% perfect for us not to notice. This means the candidates for the thread are likely to be secondary or supporting cast and thus have everyone assume it’s their natural accent since we probably don’t know much about 'em.
But if you have any examples of very well done accents in movies, let’s have 'em!
The fact that Jamie “Apollo” Bamber, from Battlestar Galactica, is British comes as a tremendous shock to anyone hearing him speak in his regular speaking voice for the first time. I have never heard of a reliable case of a person who could tell he wasn’t American from his lines on the show.
Hugh Laurie in House. I don’t think many people knew that he wasn’t from the US until told and even then it’s exceedingly rare for him to slip up on camera. I believe even the casting director, upon seeing his tape, is said to have been glad to have found an American for the role (or something like that).
Both Russell Crowe and Guy Pierce in LA Confidential.
Two Aussies playing 1950’s LA cops.
Sam Neill (a native New Zealander) does an excellent American accent.
Renée Zellweger in either of the Bridget Jones films.
Gwyneth Paltrow in Shakespeare in Love.
The number of non-US actors and particularly actresses doing excellent American accents on television and in films these days is surprisingly large. In addition to those already mentioned there’s also Poppy Montgomery (Unforgettable), Yvonne Strahovski (Chuck), Sophia Myles (Moonlight), Simon Baker (The Mentalist), Kevin McKidd (Grey’s Anatomy) and probably a bunch of others I’m forgetting. Cate Blanchett and Nicole Kidman have done good American accents. My favorite though is still Tracy Ullman. She has done American accents in film and on television. I was most impressed many years ago when she did a sketch on her show entirely in a spot-on Baltimore accent. I’ve never heard that done before or since by anyone, American or foreign.
It’s harder for Americans to judge how good foreign accents are being done, but Meryl Streep has never been criticized (as far as I know) for the many she’s attempted. Gwyneth Paltrow’s English accent is also quite good.
E.T.A. I was too slow on the Gwyneth Paltrow. I think she also did an English accent in Sliding Doors.
Toni Collette’s Philadelphia accent in The Sixth Sense. Absolutely flawless.
Gordon Jackson’s speech was so intelligible, even in a loud racket, my mother said he was the world’s best English speaker.
A bit more subdued was Lawrence Olivier’s narration of “The World at War” series.
My Zimbawean friends said Leonardo DiCaprio’s Rhody accent was perfect in Blood Diamonds
Really? I can’t usually tell if an American accent is real or not, but his often sounds obviously fake to me. Notably in Alcatraz.
Cary Elwes was on Leverage last night, and his US accent on that was pretty good, I thought.
Having recently watched Spider-Man, I’d say Andrew Garfield is very good. I can’t remember how he did in the Social Network.
I never would have realized Hugh Laurie from House was not American, and I never would have realized that Rick, the main character in The Walking Dead, was not only not Southern, but not American, either.
Jamie Bamber fooled me too.
Ooh, good one!
Toni Collette is fantastic. She’s Australian, and aside from doing a convincing American accent in several movies, she’s done English accents (e.g. About a Boy) that sounded fine to my American ears AND in Velvet Goldmine even pulled off the challenge of doing an American who slides in and out of an affected posh English accent.
Thinking of people who were in Velvet Goldmine, Christian Bale’s American accent is good enough that I thought he was an American for years and was impressed by how well he did an English accent in VG! But I believe he grew up partially in the US, so he may have picked up an American accent pretty early on.
In the same way, I was impressed with Gillian Anderson’s English accent, right until I read she grew up in London.
Rachel Griffiths in Six Feet Under and Mel Gibson in most of his films. Both Australians.
I’m not too impressed by Gibson’s American accent, as he has spent a heck of time in the US, but he did a flawless English accent in The Beaver despite never living in the UK.
Stayed up late the other night watching “Secondhand Lions” which is one of my favorite films. Micheal Caine’s Texas accent in that film is spot freaking ON! 
I didn’t expect him to pull off the accent as well as he did, to be honest.