Since you are traveling in cherry blossom season, then I'd also recommend you start from the south and work your way up. While the official cherry blossom season is forecast to start March 22nd in Fukuoka, the peak doesn't arrive until about a week later. Mother Nature can also be quite fickle. There are also different varieties of blossoms that don't all bloom at the same time.
Ohori Park though has next to no cherry blossoms worth mentioning. But next to it is Maizuru Park with the castle ruins, which is one of the best places in the city:
Ohori is worth seeing and taking a stroll however,
and there is another wonderful cherry blossom park just 10 minutes on foot north of it - Nishi Park:
There is also Minami Park for blossoms as well as the zoo and botanical garden, but the above 2 are the best. For some other good info to see on visiting Fukuoka, try:
Dazaifu is not the place to go for cherry blossoms though. It is famous for plum blossoms, which come out earlier and are actually in bloom right now. They may be gone by the time you get to Japan.
Kyoto will be extremely busy during cherry blossom season - be sure to take at least half a day and go see Nara as well.
The best place for blossoms in Japan is on Mt. Yoshino in Nara - they are forecast to start on April 3rd - if you can go there a bit after that you will be rewarded with a mountain with over 30,000 cherry blossom trees. For Kyoto spots, see:
Your best chance to see the blossoms might just be in Tokyo if you'll be there the first week of April. The best places to see them are Ueno Park, Shinjuku Gyoen Park, Nakameguro and Yoyogi Park (near Harajuku & Meiji Shrine).