The first we did recently was go to the ruins of Fukuoka castle coincidently the week after we went with Eiji and Erika! Within the grounds of the castle (which is now a huge park) is an archaeological dig and small museum of the Fukuoka Korokan which was a guest house for foreign emissaries and merchants from the 7th to 11th century. The remains were only found in 1987 and they're still digging them up. Apparently there were some in Kyoto and Osaka but only in Fukuoka has remains been found.
After having a gander at that we wandered over to the castle ruins where all the plum blossoms were coming out. The area was once again filled with Japanese people with huge cameras. Funnily enough there was one Japanese guy who was taking pictures of the blossom with his phone who began talking to Hirose-sensei, and to me in broken English until he realised I could understand him better in Japanese. He then proceeded to take me around the area showing me and Hirose the different types of trees and what shots of the blossoms would make good pictures. Apparently there were 16-18 different kinds of plum tree in the park and they originally came from China.