The third of the four things that the early disciples devoted themselves to was "breaking bread." The second half of 1 Corinthians 11, which discusses the problems caused when some people ate all the food without waiting for others to arrive or even got drunk (!), makes it plain that this was in the context of a full meal (verse 21). It isn't referring to a fragment of cracker or bread and a thimbleful of wine or grape juice taken solemnly and silently together! We know too from Acts 2:46 that the new believers shared their meals together.
Most simple/organic churches meet in the context of a meal. There is something about eating together that enables fellowship, and it's harder to be "religious" where food is involved. Eating together usually involves laughter and sharing, good-natured banter and deep heart-to-heart discussions. As one of our friends likes to say, "How do you spell fellowship? It's four letters:
Most groups that we know share a potluck meal–it is reproducible and doesn't leave too much work with the host family. A lot of fellowship goes on too over the preparation of food and the clean-up later. Some groups may even have their whole time together around the dining table.
What about communion. We often add taking the bread and wine in remembrance of Jesus' death on the cross into the context of the meal. Maybe we break into smaller groups to share together, or have each person share with someone they would like to pray for. The Lord is very creative and again, and if we avoid repetitious practices, it doesn't become a religious ritual.