"Traditional" music is a tough one. There are several different kinds of music one might call "traditional" but only some of it is really accessible.
Classical: There are a few Japanese classical composers where the "Japanese-ness" shines through - Ifukube, Takemitsu, and Wakasugi come to mind. Classical music by these composers is not really traditional though - you'd be hard-pressed to find a Japanese person on the street who'd ever heard of any of them.
Pop: Modern J-Pop is a real toss-up. Some of it is grating in a way that most westerners could not imagine possible. Some of it is better. Some of the older pop (50s) music could be considered traditional.
Enka: This is "traditional pop" - male or female soloists singing somewhat shmaltzy songs with somewhat shmaltzy orchestral backing. Imagine the cheesiest C&W and slow blues love songs you know, converted into elevator music, with over-dramatized singing over the top of the massed strings, done with superb control of both vibrato and melisma. That's enka. Maybe that's a bit harsh (though maybe not - it is an acquired taste). It IS however likely the only accessible modern Japanese music using the Japanese pentatonic scale - in use since before Chinese and Korean influences. Though there are Chinese and Korean versions of enka, and in modern times, it is not clear which one has influenced which more.
A list of "traditional" Japanese instruments which are reasonably well-known in the west would include the koto, the shamisen, the shakuhachi, and taiko drums. It is probably easy to find "sampler discs" of music of any of these instruments on Amazon but I will try to come up with a good idea for shakuhachi. I know a guy who knows a guy...
Truly traditional Japanese music would be Japanese imperial court music, called "gagaku", which has its origins some 1200-1400 years ago (and which probably hasn't changed much in the last thousand years). The musical instruments used include the sho, the ryuteki, the hichiriki, the komabue, the biwa, the koto (and it's lesser-known cousin, the wagon), the kotsuzumi (and the larger okawa), maybe even the shakubyoshi, and probably several others which I've forgotten. Other traditional Japanese music would be "nohgaku" (music to accompany noh plays), biwagaku (unaccompanied biwa), sokkyoku (which is koto, shamisen, and shakuhachi together), and Okinawan music using the sanshin (a version of the shamisen), among others.
Enjoy!