I had this same experience when I moved from Seattle to rural Iliinois a few years ago. Two of my neighbors invited me to their church, and another neighbor is a church. Luckily, my MIL who lived here (and is the reason we moved here) was a member of a different church, so I could beg off by mentioning that and avoid the whole atheism discussion completely.

To this day, I've never mentioned it to any of my neighbors.