Tchaikovsky’s Slavonic March, particularly the first few minutes, has become one of the foremost “Russian sounding” pieces used in documentaries and movies (to the point of cliche IMO, though I like the piece). It uses a number of Russian and Serbian folk melodies.
I doubt it’s this, but the finale of the 1812 overture includes the (anachronistic) Russian national anthem. God knows it’s one of the most famous pieces in the history of ever.
As you already may have guessed, my knowledge of classical music is extremely poor.
It is none of the above, so I’m thinking the Russian angle may not be correct. It’s not an instrumental version of something as I’ve never heard it with lyrics! It’s always just orchestral, mainly strings. It is as famous as the 1812 Overture. Hell, I’ve heard it so many times, I just can’t remember where exactly.
I’m going to try one of those websites that you key in the riff. I’ve tried before and had no luck but will try again.