An island which most Nagasaki locals have never visited, I'm lucky enough to have been 7 times in 3 years. While it's a bit hard to get to (by air or by ferry from Karatsu, Saga or Fukuoka City, Fukuoka), it's worth the effort.
Iki is well known by the Japanese for its uni (sea urchin) and shochu (rice whiskey), foreigners are usually more attracted to the stunning beaches.
There are a few main beaches on the island including Ohama, Twins, Nishiki, Tsutsukihama. Every one has its own claim to fame, but the most popular among tourists is Nishiki, for it's marine sports area where you can get rides on jet skis, banana boats, rent snorkeling equipment and more. There is also a bar with drinks and sweets and beach volleyball nets.
Locals frequent Twins and Tsutsukihama beaches, which are often less crowded. Tsutsukihama also has a camp site and bbq area.
Transportation around with island is a bit tricky, so if you don't bring a car on the ferry, you might want to think about renting one or looking into accommodations with shuttle services.
Also keep in mind that the downtown areas are small and often close early.
Iki is definitely out of the way if you're just traveling around Kyushu, but if you are looking for a beach getaway, this is a beautiful, secluded option.