The construction zone that was our house for 5 months, during the sunroom addition, answered the question for me. All of the dust certainly did no favors for the wear & tear of the chairs... So recovering was a go!
I hit up Mary Jo's (one of my favorite weekend spots) and found this perfect-shade-of-gray linen fabric for the chairs.
We unscrewed the base of the seat from the frame, cut pieces of batting to the same size as the seat, and cut the fabric a few extra inches wider than the seat (allowing for enough extra fabric to staple down).
Then we laid the batting over the top of each seat, covered with the fabric, and stapled to the bottom with a staple gun (with one of us pulling the fabric tightly and the other stapling down). We stapled once around the seat, then went back and stapled even closer together.
For the corners, we folded the edges (like wrapping a present) so that the front would appear flat and the only folded edge would be off to the side.
Then we cut away any excess fabric and re-attached the seats. For a refreshed final result...