The original Fukuoka Tower, since overshadowed by the newer Tower in Momochi.
Hakata Port Tower.
Still retains a "retro" charm. Maintained by the Hakata Port authorities, this would have dominated the old Fukuoka docks. Good views of the city and over the bay from the top - and it's free. Small "history of the Tower/docks" display at the bottom, though sadly in Japanese only. The Tower is 103m high. See the "Hakata Port" link for more info.
Looking west over the harbour and Atago bridge, with Momochi in the distance.
It is possible to walk from Tenjin (head for the coast, turn right down Nanotsu-dori, then follow the signs). It is about a mile. Otherwise bus or taxi is best although it is possible to walk from Gofuku-machi subway station if you are moderately masochistic.