Probably most of the foreign tourists visiting Japan will purchase and use Japan Rail Pass. However, Japan Rail Pass cannot be always the cheapest way to travel Japan. It can be cheaper but may be you are wasting your precious time in Japan, spending more times on trains rather than having a sightseeing.
Then, in case you want to go to Fukuoka in Kyushu from Tokyo, flight ticket can be much cheaper than taking Shinkansen. Also the traveling time including waiting time at the airport is much shorter when taking a flight. Shinkansen will take at least 6 hours from Tokyo to Fukuoka by Japan Rail Pass holder, including one transfer at Shin-Osaka or Shin-Kobe station. Flight time from Tokyo Haneda to Fukuoka is less than 2 hours. Even adding 2 hours for traveling to and from airports, the total hours is 4 hours. Two hours less than Shinkansen.
Then, if you want to travel to city in Kyushu from Fukuoka, probably buying a discounted ticket online is cheaper solution. For example, JR Kyushu offers online reservation for their limited expresses running on Kyushu. Kyushu Net Ticket allows you to reserve the seat if you select reserved car and transaction is done by credit card. Unfortunately you cannot board the train by the confirmation email but you have to do is to visit the ticket office or the designated ticket vending machine before the train's departure. You can book the ticket up to 6 minutes before departure and also able to modify the booking too.
Ordinary fare from Hakata station in Fukuoka to Yanagigaura in Usa, Oita prefecture is JPY 2530. However, Kyushu Net Ticket is just JPY 2020. Also you will get a discount for limited express supplement. Save money and spend the money for dining or souvenirs!