Korean Air has one-way flights from Dallas (DFW) to Fukuoka (FUK) for $475 with a connection. Flights in August. Departures on Thursday, Friday, and Saturday.
What good are “one-way” fares though? Well, if you have 35,000 American Airlines miles, you can book the return using that out of Haneda, turning this into a more interesting Open-jaw itinerary.
An “Open-Jaw” is when your destination airport isn’t the same one you’re flying out of. In this case, fly into Fukuoka, then build your itinerary toward your exit airport (Haneda). It’s fairly easy with Japanese rail and you’ll be able to visit more of the country during your trip without backtracking.
Here’s a more detailed example dates for this: Fly out of DFW into Fukuoka on August 27. Buy (with miles/points) the AA flight from HND to DFW on Sep 03 [link]. Spend a few days in Fukuoka/Hiroshima, take the bullet train to Osaka/Kyoto/Kobe, spend a few days there, take the bullet train to Yokohama, catch your flight out of Tokyo on September 03. Adjust your days you want to stay in each city, or select a return flight further out if you want longer than a week.