So, I took the chords from the live version of this song, played by Marcus Mumford 
and Oscar Isaac at the Caffe Vivaldi (http://www.youtube.com/watch?v=s8q4HfvwAT4). 
I guess the version of the movie "Inside Llewyn Davis" will be pretty similar to this one.

- Open D tuning [D A D F# A D]. You can use A standard tuning too.

Chords:
D  [000000]
G  [500500]
Bm [900800]
A  [700700]


D                   G      D                  
If I had wings like Noah's dove
            Bm           G      D                      
I'd fly the river to the one that I love
D               G   D      A         D           
Fare thee well, my honey, fare thee well


D                    G        D              
I had a man, who was long and tall
             Bm          G      D                 
He moved his body like a cannon ball
D               G   D     A          D                                     
Fare thee well, my honey, fare thee well

D                             G       D     
I remember one evening in the pouring rain
          Bm                 G      D       
And in my heart there was an aching pain
D               G   D     A          D          
Fare thee well, my honey, fare thee well

D                G         D        
Muddy river runs muddy and wild
               Bm              G      D       
You can't give a bloody for my unborn child
D               G   D     A          D      
Fare thee well, my honey, fare thee well

D                             G      D      
Just as sure as the birds are flying high
                Bm                      G           D       
Life ain't worth living if you ain't with the man you love
D               G  D      A          D       
Fare thee well, my honey, fare thee well


Fare thee well, my honey, fare thee well