We are a Christian family. I have a curriculum for their Bible Lessons. Again I have this but it isn't always used we go with the flow mostly. Anyway in this curriculum they have memory verses. I don't really make the boys memorize them though. Why? Well I guess I feel the Bible should be experienced and not necessarily memorized. Sometimes (I know it would with me) we get so preoccupied with memorizing we forget to look at the meaning and feeling of what the Bible is saying.
The boys a few months back for Church (Potters Kids) had memorized verses. I helped them with it but it was done in a fun game way and I never forced them to learn them. They may not know a lot of verses word for word but they do know what the Bible has to say and know most of the stories in the Bible.
We read the Bible as a family most daily (I have to admit I do forget at times. Bad mommytimers I tell you LOL (Mommytimers - like Alzheimer's only you get it when you become a mom LOL). So it's not as if we are not learning from the Bible but we experience it as a family and enjoy reading it as a family. It doesn't become work for us but a family time. This is why we don't memorize Bible Verses.