The River is written in the key of B Dorian.
According to the Theorytab database, it is the 6th most popular key among Dorian keys and the 35th most popular among all keys. The B Dorian scale is similar to the B Minor scale except that its 6th note is a half step higher (G♯). Chord progressions in Dorian have a characteristic sound due to the major quality of the chord built on the 4th scale degree.
See the B Dorian Cheat Sheet for popular chords, chord progressions, downloadable midi files and more!