I tend to take it on a case by case basis. Many kids movies are PG these days even though they are much tamer than the G rated versions from my childhood. Some PG-13 movies are also kid friendly, Adams Family and Doctor DoLittle are two that come to mind. It really depends on the movie and the reason behind the rating for me. R is not a rating I feel comfortable with yet though.
As for the Hunger Games, I read the first book and I wouldn't let my dd see it. I told her what it was about and she decided she would rather not see it anyways so it wasn't an issue for us. I was surprised to learn that the books were targeted towards kids, but I think they mean kids meaning teenagers not kids meaning elementary school children.