If you don't mind, I concentrate on music featured in Russian films. I had so many choices that I had to sort the movies of my choice by timeline of production. If you can't understand Russian, just enjoy the sound of it.
Легко на сердце от песни весёлой (A Happy Song) from Весёлые ребята (Jolly Fellows), 1934
This song is best known as the March of Stalin's Tankmen,
best watch starting from 2:15 in this clip - from Трактористы (Tractor Drivers), 1939
Тёмная ночь (A Dark Night) - from Два бойца (Two Soldiers), 1943
Здесь вам не равнина (You are not in flatlands) - from Вертикаль (The Vertical), 1967
Когда усталая подлодка... (When a tired submarine comes home) - from Командир счастливой "Щуки" (Commander of a lucky
Schuka), 1972
Opening titles music from Soviet version of
The Adventures of Sherlock Holmes and Dr. Watson, 1980
Прекрасное далёко (A beautiful faraway) - from Гостья из будущего (Guest from the Future), 1985
Перемен! (Change!) - from Асса (Assa), 1987 -
the song actually begins from 1:00
Bonus: A folk song which was popular during the WWII "На поле танки грохотали" (The Tanks were rattling like a Thunder), which was also featured in a movie, is sung in English by some American students. They manage to get it atmospheric.
[ame="http://www.youtube.com/watch?v=xeb3ZLb-wJ4"]http://www.youtube.com/watch?v=xeb3ZLb-wJ4[/ame]