The Nihon Sankei are the Three Views of Japan, a designation dating back to 1643. They are Miyajima, Amanohashidate, and Matsushima. Don't expect majestic scenery — the beauty is much more subtle.
Miyajima is famous for the floating red torii gate of Itsukushima Shrine. Depending on the tide, you might find it standing in water or lodged in the mud. You reach the island by ferry from Miyajima-guchi, on the local line from Hiroshima.
Amanohashidate is a pine tree-lined sand spit in northern Kyoto Prefecture. Tradition says you should view it by bending over and looking through your legs — the "bridge to heaven" then appears to float in the sky. You can get there by train from Kyoto station, though JR Rail Pass holders should note there is an additional supplement for the private leg of the journey.
Matsushima is a collection of pine-covered islets dotting a bay near Sendai. Take the local train from Sendai to Matsushima Kaigan station, then walk along the waterfront or take a boat cruise through the islands.