I think aesthetically, the woven look is lovely in the overall quilt design. It's not too much at all, and really enhances the applique blocks.
Yet another quilt with solid alternating blocks? (Sorry, but that's what I think everytime I see one.) ;-) So I'm all for doing something in those blocks, and the cream/white is so subtle, it's almost just adding a bit of texture.
What size are would they be in your layout?
Are you totally against just piecing the blocks to get the same effect, as you really want to "weave" the strips?
And apologies if you already know this, but if you piece it, then those green squares are just setting triangles (there are a gazillion pages out there with the details).
I don't think the look would be different. Is this going to be a used quilt or a wall hanging? Because there are some finishing things you'd have to do to the strips to make the block more sturdy (so the ends don't fray, and they don't turn into crumb catchers) if it's going to be handled.
The applique blocks are also lovely. Are they original?