I second Get Smart's recommendation of Kobo Abe (I translated and commented extensively on this author for my senior thesis in college). In particular, his Woman in the Dunes
is one of the modern classics of Japanese literature. It was also made into a brilliant movie.
Mishima is also good but was a better critic than author. He wrote an excellent story about seppuku
(ritual suicide by disembowelment) as aesthetic ideal. I don't recall the title at the moment.
Yasunari Kawabata's Snow Country
is wonderful and subtle. He won the Nobel Prize for Literature for this, largely thanks to Edward Seidensticker's translation.
Jun'ichi Tanizaki is also outstanding and quite accessible. In particular, In Praise of Shadows
is a wonderful short essay on Japanese aesthetics.
Ryunosuke Akutagawa is also worth reading, though I can't recommend one work in particular. His progressive madness culminating in suicide precipitated a kind of cultural crisis.
Anything by Natsume Soseki is worth reading. He is arguably the first Japanese author of the modern age.The Book of Tea
by Kakuzo Okakura was originally written in English and is a delightful read on the subject of Japanese aesthetics.Ichiyo Higuchi
was a Meiji period authoress who is definitely worth a read. I have only read her Thirteenth Night
, which I don't believe has been translated. Her likeness now graces the 5000 yen note.The Narrow Road to the Deep North
by Matsuo Basho is the most read piece of classical Japanese literature and is a must-read if only because it was written by the father of haiku.
I also second Brian SD's recommendation of the Kodansha series with the facing-page translations.
For a medieval war story, The Tale of the Heike
is a really amazing epic. This is the seminal samurai drama.
If you're feeling particularly ambitious, read The Tale of Genji
all the way through. This is the mother of all Japanese liturature. I have not read the Seidensticker translation (shame on me!), but have read the more recent Royall Tyler translation, which is perhaps better for students of classical Japanese liturature, as Tyler has done his best to approximate the flavor the syntax of the original.