I've recently gotten seriously into this type of music, and I thought I'd recommend some of the bands and albums I've heard.
Wardruna -- Shamanic Neo Folk band from Norway with an ambient and archaic touch. Heavily focused on the use of old traditional Norse instruments such as keyed fiddles, horns and Norse shamanic drums, as well as chants. Lyrics in Norwegian.
Tenhi -- Neo Folk band from Finland with acoustic guitars, flutes and other traditional instruments, accompanied by a rock drum rhythm. Lyrics in Finnish.
Vali -- Norwegian Neo Folk with acoustic guitars and other traditional instruments. Instrumental.
Nest -- Ambient Neo Folk from Finland with acoustic guitars, kantele, shamanic drums and occasional electric guitars. Lyrics in English.
Neun Welten -- Neo Folk band from Germany. Similar to Tenhi and Vali. Lyrics in German.
October Falls -- Finnish Neo Folk, as well as Black Metal on some albums. Similar to Vali.
And some Neo Folk albums released by Black Metal bands are:
Kveldssanger by Norwegian Folk Black Metal band Ulver. Similar to Vali. Lyrics in Norwegian.
Urminnes Havd (The Forest Sessions) by Swedish Viking Metal band Manegarm. Lyrics in Swedish.
Origin by Norwegian Black Metal band Borknagar. Perfectly blends Neo Folk music with Progressive Rock. Lyrics in English.
Visor om Slutet by Finnish Troll/Folk Metal band Finntroll. Lyrics in Swedish.
Feel free to suggest other bands if you happen to know any good Neo Folk, or other Folk related bands/music.