I just discovered Christian Cinema, a Netflix-type company that only carries family and faith promoting movies!
Since then I discovered another source of Christian movies: Christian Film Database. It appears that you can rent or watch movies using pay-per-view. They promote films with a Christian message, and claim to be the most extensive database of Christian movies ever.
Zion Tube is a great movie resource with an LDS perspective. Created by a Utah homeschool family, it highlights heroes and heroins from history, carries movies about history, government, religion, science, and so on. Some are full-length movies, some are clips, and some are trailers to help you see if you want to rent or buy the movie. Great site! (Sorry, I don't remember the name of the family who created it or I would give them credit!)
Clear Play DVD Player Filter
The ClearPlay DVD player filters out inappropriate parts of DVDs according to how you program it. It can filter out swear words, a bedroom scene, violence, and so on. It is not a cure-all. It cannot filter out a dark theme, or the fact that two people are living together without being married, or things like that. The customer purchases a monthly membership to download the current and past movie filters from their website. We purchased a short-term membership and then cancelled it. That means we don't have access to the filters for brand new movies, but that's okay. We don't watch movies very often, and when we do, we usually just choose something that's clean to begin with.