I picked up a bike in Chinon, rode through Villandry to Amboise, then around the Chateaux in that area. Overall, it worked pretty well. I lost the bike path through Tours and ended up on fairly narrow highways with lots of truck traffic, a little more adventure than I had expected.
Chinon isn't served by the main bike route, so I had to follow highways (mostly comfortable riding though) until I got to Villandry. It's also considerably more hilly between Chinon and Villandry than most of the bike pathed area. The initial climb out of Amboise toward Chenonceau wasn't really "rolling" hills, after the initial climb the rest of the ride is fairly comfortable.
My advice is plan your route a little more carefully than I did. Know the route rather than relying on a GPS for guidance. I used CoPilot with my phone for navigation. It worked fine with my car but horribly in bike mode. When I was on the bike path, it was either because I knew where it was ahead of time or stumbled across it. CoPilot didn't seem to know a bike path existed. It also took me down a dirt road, between Chinon and Villandry, that dead-ended probably ten miles into a forest. Rather than backtrack, I opted to carry my bike (and bag) and forge ahead. I probably went two or three miles, over hills and across a stream, before I picked up a road and could start riding again. It was a sunny day, so I knew which direction I wanted to go.