Going the other direction is a bit more difficult. So far the best US cover of an RU song I've run across has been Chet Atkins:
Atkins - Очи чёрные
I'm sure you all are much more color-blind, and it's just my old-fashioned XX upbringing that caused me some cognitive dissonance in the previous card's collection of pale guys singing "go down moses", but judging by the parodies, russian radio also tends to segment by audience ethnicity:
БОНЯ И КУЗЬМИЧ - ЗА МОЛОКОМ
on the other hand, although russians definitely have fewer integrated acts than western europe, in certain cases it seems to be due to lack of opportunity rather than lack of inclination:
Валенки.Марина Девятова и Пьер Нарцисс.
MC Doni feat. Натали - Ты такой