Skynet Asia Airlines has announced it will introduce some discount fares for foreign visitors who can show a short term visa stamp in their passports. The deal will be offered on 8 routes including Haneda-Miyazaki, Kumamoto, Nagasaki and Kagoshima and between Naha (in Okinawa) and the same 4 Kyushu cities. You can't book a flight in advance: visitors will have to turn up on the day and see if seats are available. All flights will be 10,000 yen (normal fares for the routes range from 13,000-31,500 yen). The deals are on offer from December 1st.
More details here (English)