Here's a great link for making your own grocery (and other stuff) bags. MAKE YOUR OWN BAGS Patterns are linked for knitting, crocheting, sewing, and even if you don't craft, there's a tutorial at the end on how to tie a furoshiki. (A Japanese carrying cloth)
All you need for the furoshiki is a square cloth of moderate to large size, and these two links ARTICLE DIAGRAMS
And the article even has some video showing how to do it.