Thanks to everyone for the great ideas. It is really helping me get a handle on this project! Still not sure what exactly I'm going to do. So far I'm not too happy with strip piecing on a foundation (plus it is annoying not to be able to iron the Vilene squares), so I am thinking of simply piecing strips together to make strip sets, cutting squares out of that, then putting them together with a gray sashing fabric.
I did find several quilts online that I really like. Here are some links to them, even though mine is likely to be considerably different from these:
http://fussycut.blogspot.com/2011/07...ing-quilt.html
http://3.bp.blogspot.com/-e2JbybIprS...full+front.jpg
and this one, which makes very clever use of blues by adding larger pieces in two corners:
https://www.pinterest.com/pin/292804413251821299/
Apparently I find the idea of complete randomness very uncomfortable, so will have to have some planning theme for my quilt. Don't want too much, though.