We will have 1 week in mid June after a tour that ends in Chamonix. Looking for best area to bicycle and enjoy beauty of French countryside and rural culture. Any recommendations?
You will find more cyclists at crazyguyonabike.com
It kind of depends on what kind of cyclist you are, and what the rest of your itinerary is like.
From Chamonix you might consider Burgundy. It is largely rural, very pretty, has great food and wine, and is well situated to continue (or fly home from) Paris. You'd have a choice of terrains.
If you are after something more challenging, you might want to stay in the Alps, or go to the Jura.
Choose carefully. While I was driving in Burgundy in mid-June, I often prayed that I would not meet any cyclists. Roads were narrow, winding, not well-maintained and without shoulders. In Alsace at least the roads were wider and in very good condition, but still no shoulders.
On your way out of Chamonix, Annecy can be a cute-town stop with a level bike path around the lake.
Another vote for beautiful Annecy!