If you're staying a bit south of Hakata Station and looking for a nice park to work out, I can recommend Morooka Koen. There's a running track 650 meters long that forms a loop around a lake. You can enjoy a nice mountain view and see the reflection of the sun in the water at sunset while you're burning off calories. Morooka Koen is close to Sasabaru Station, two stops from Hakata Station.
Experience the oceans like never before