Well I already weighed in on padded handles, but let me add another idea: mesh.
One of my favorite farmer's market bags for a long time was a mesh backpack made by one of the big producers. Mesh is good for a lot of kinds of produce and stuff that likes to breathe. Also, it renders the finished product light and easily compacted into a small bundle.
Personally, I would always take a bag like the lux for heavy stuff, and carry the other bags inside it.