Notoriously, Sean Connery’s BAFTA nominated performance in The Hunt for Red October, came with no actual attempt at a foreign accent.
(Sean Connery as Marko Ramius in The Hunt for Red October.)
For the record it wasn’t supposed to be a Russian accent he was using to begin with. The character of Marko Ramius is Lithuanian, and his nationality, ethnicity and “foreignness” to the vast majority of Russians he finds himself in the company of, is a major element of the plot and of his character and motivations.
The accents are quite different, and playing up the contrast between the Lithuanian Ramius and his Russian crew was supposed to be intentional.
Connery, however, was a last minute replacement for Klaus Maria Brandauer. He had only one day to prepare or rehearse for the part. In that day, he managed to learn all his lines, including lines in Russian.
That, in itself, is no small feat, particularly given he was leery of taking the part to begin with – he had good reason to believe the public was not interested in Cold War thrillers anymore.
Regardless, Connery himself always admitted he struggled with accents. And with no time at all to prepare, there wasn’t much he could do to learn a relatively uncommon accent like Lithuanian.
So rather than try for a ham-fisted, stereotypical and vaguely Slavic accent, he essentially just used his own Edinburgh accent.
It actually did it’s intended job, as well.
(Tim Curry as medical officer, Dr. Yevgeni Petrov in The Hunt for Red October. Curry made a real attempt to nail a natural sounding Russian accent. Whether he succeeded or not is up to the viewer – but there is a clear contrast between his accent in particular and Connery’s.)
Ramius, a Lithuanian, was always supposed to sound different and stand out like a sore thumb from his overwhelmingly Russian crew. So Connery actually managed to convey his character’s “foreignness” simply by sticking to his natural voice.
It turned out to be one fantastic performance, as well.