Some do, some don't.

Some Mormons will tell you that past presidents of the Church have commanded us to not watch R-rated movies. As near as I can tell, this is simply not true.

The source of the confusion seems to be this quotation from Ezra Taft Benson, president of the church in the 1980s:

"We counsel you, young men, not to pollute your minds with such degrading matter, for the mind through which this filth passes is never the same afterwards. Don't see R-rated movies or vulgar videos or participate in any entertainment that is immoral, suggestive, or pornographic. Don't listen to music that is degrading."

(from April 1986 General Conference, also found in the book, "The Teachings of Ezra Taft Benson", under the "Priesthood" section)

Note that he is specifically talking to the youth of the church.

To reiterate, I have never seen/heard church leaders tell the adult members of the church to not watch R-rated movies. I personally avoid R-rated movies, as a protest against the movie industry ruining many perfectly good films by including unnecessary sex, violence, and gore.