Shibuya Ward has created an excellent place in Yoyogi Park for dog owners. A large section inside the park has been set aside so that owners can play with their dogs and participate in many dog-related events. I have uploaded a photo of the area and as you can see, it is pretty well shaded. It’s also a great place to meet other dog lovers. Below is a video of the area.
Here is a site with information. It’s pretty good with photos, but it’s only in Japanese→ http://ke.cool.ne.jp/dogrun/
You have to register to use the space by calling 03-3409-6081. The location is in central field of the park, toward the cycling center. Here is a map → Yoyogi Park If you look at the map, the location will be near the toilet in the upper left corner of central field. Enjoy!