I have met most of my friends through church. I have a board game group that meets every Monday night (best night for all of us) and everyone there first got to know each other at church.
I have been tempted to look into this, I'll admit. I remember when I used to do Sunday youth group every week as a kid. If I could find a local one that just had a regular board game night or something I'd probably be down to give it a try.
If you find a traditional church to be a bit too heavy-handed on some current wedge issues for your liking, you might find a Unitarian Universalist church to be more like what you describe.
What about a church? They do the weekly hang pretty much every Sunday.