Kirk here. In a separate post, Joe mentioned a friend who was interested in cycling:
https://www.facebook.com/Kumamotoi/posts/2892183834188363?__tn__=-R
This gives me an opportunity to collate and reintroduce some old information that I’ve posted previously about the Yuka Family Road, a cycling road.
First, the post I’m sharing has a YouTube video that will give you a taste of what cycling on the road is like. Also, you can see a video of the full route (!!) with Japanese narration here:
http://youtu.be/Hw8MyeAE0oA?t=50m14s
Second, are some maps – a pdf of a map to which I added some English ages ago and a digital (though not Google) map:
http://www2.kumagaku.ac.jp/teacher/~masden/kumamoto-i/yuka_cycling_map.pdf
https://www.alltrails.com/explore/recording/--10322898
Third, here’s a post about a venerable old tree that you can visit by taking a short detour from the cycling route:
https://www.facebook.com/Kumamotoi/posts/2635849163155166
Fourth, here’s a page on Wikiloc.com with information about other biking possibilities in Kumamoto. It doesn’t mention the Yuka Family Road, however:
https://www.wikiloc.com/trails/cycling/japan/kumamoto
Hope this helps!