DVDFab has a solution for some Blu-ray movies, to remove Cinavia. But as far as I am aware, they haven't extended that to DVD's yet.
With the present solution, you'd need to start with an original Blu-ray disc from the studio, then use the DVDFab Cinavia removal module in combination with their Blu-ray Copy or Ripper sections. The output from these would be a Blu-ray movie, or a movie file, like an MP4 or MKV.
If most of your movies are now in DVD format, I'd stick to using a DVD player for them, since a DVD player will not have Cinavia detection.
Another solution, which is proving very popular, is to use a media streaming box, like a WD TV Live or a Roku, and play from them. I normally use MKV files in my WD TV Live.