“Folk Japanese music”: Which?

Min'yō: Folksongs, often working songs, shanties etc.

Koten-uta: A generic word for “old songs”. Often used for “city songs” (miyako-bushi) like Sakura Sakura, Edo-period koto compositions and the like.

Jiuta: Edo-period, urban ensembles of koto + shamisen + shakuhachi.

Warabe-uta: Children's folksongs.

Komori-uta: Lullabies, a surprisingly productive genre.

Tsugaru-jamisen: A distinctively fast shamisen genre from the Tsugaru peninsula, kinda power metal shredding on the shami.

Gagaku: Classical/courtly music, based on Chinese classical music.

Ohayashi: Festival songs. Drums, whistles, flute loops, yells.

Hayashi/noh theatre music: Not to be confused with the above. Ghostly and eerie. Dissonant melody on purpose, no harmony, no conductor.

Kabuki theatre music/nagauta: A “mainstream” counterpart to the opaque noh music.

Biwa-uta: Epic poetry played with a lute (biwa).

Shōmyō: Buddhist chanting. Excellent if you like droning.

Honkyoku: Shakuhachi pieces, solo or (rarely) in two or more. No set rhythm; follows the breath and intuition of the performer. Lots of note-bending, breathiness, rough sounds for effect. Sounds like what bamboo would play if bamboo could play music without humans involved. Originally Buddhist music, was forcibly secularised.

Kagura: There's two distinct styles for holy shintō music; mi-kagura, peformed at the imperial palace, is close to gagaku, while sato-kagura, performed for dances and folk rituals, is closer to ohayashi, with fast flutes and accelerandos.

Jojōka: “Lyric songs”, a sort of Meiji-period folk revival, blending a reevaluation of Japanese musicality with Western concepts of music. Set to authoral poetry. In particular, school songs (shōka) acquired a bit of the status of folk songs due to their use in education (Kōjō no Tsuki is in this category). Also includes kayō—at this point we're at pop songs from the 1910s/20s.

Enka: “Ballads”. Something akin to country or blues, characterised by pentatonic scales, very heavy vibrato, and sad lyrics about lost love and drinking. Stereotypical “karaoke music”. Stereotypical “old person thing”.

Taiko: Big chonky satisfying traditional drums, but played with jazz drum technique in large ensembles. See it live if you have the chance, it loses a lot on recording. Newer genre than people think.

Ryūkyū: The archipelago is a distinct culture colonised by Japan and has its own entire musical tradition, that I haven't looked into yet so I won't try to list.