Carmike Cinemas Summer Kid's Series will feature $1 movies, $1 drinks, and $1 popcorn. Click HERE for the complete schedule of movies and more info.
Regal Cinemas FREE Family Film Festival is every Tuesday and Thursday at 10AM and is completely FREE. Last year kids popcorn and drinks were a $1 each. It isn't listed anywhere that I can find on the net so it may differ per location. More info HERE.
Malco Roxy Theatres is having their Summer Kids Film Fest. Admission is $2 on Tuesdays and Wednesdays at 10AM. Click HERE for more info.
Regal seems to be the best bet on cheapness but all are a real steal if you look at how expensive the movies are these days. From my past experience you will need to arrive EARLY!!! I went one time and barely found seats for me and Avery.