I'd like to recommend the lovely, cool, meandering, groovy 'The River' by Terry Reid.
I've been planning a 'river' song compilation: here are some others I've thought of:
2 songs by Randy Newman: 'Louisiana 1927' and 'Burn On, Big River' from 'Good Old Boys'
A song on John Martyn's 'Bless the Weather' called something like 'Back Down the River'
'Yes, the River Knows' by the Doors (Waiting For the Sun)
Many Rivers to Cross - Jimmy Cliff
Train and the River, by Jimmy Giuffre
I cast my net wide and include 'The Rio Grande', a great jazzy choral piece by Constant Lambert from a Sachaverell Sitwell poem (1920s), and Schubert's Die Forelle (the Trout, the song, or the adaptation of its melody in the eponymous piano quintet).

Why not?