This isn't correct if you're trying to sing along with the album version, but I've 
been covering this song with these chords for a while now and it sounds great.

Capo 4 (All chords relative to capo)

Intro (repeat 2X): C, G, D, Em (hammer on/off of the "a" string)

Verse I:

C
        G           D               Em      C
I'm not sure why it always goes downhill
           G         D               Em     
Why broken cisterns never could stay filled
            C
I've spent

Chorus I:

    G             D       Em
Ten years singing gravity away
        C               G               D     Em     
But the water keeps on falling from the sky

Verse II:

C
         G                D                  Em     C
So here tonight while the stars are blacking out
           G                   D           Em
With every hope and dream I've ever had in doubt
             C
I've spent

Chorus II:
     
    G               D                   Em
Ten years trying to sing these doubts away
        C              G               D      
But the water keeps on falling from my eyes
    C      G      D      Em       C
And heaven knows, heaven knows
                  G            D
I tried to find a cure for the pain
C     G        D               Em     C        
Oh my Lord, to suffer like you do
              G      D     C
It would be a lie to run away

G, D, Em, C

Verse III:

            G    D                   Em       C
So blood is fire pulsing through our veins
             G          D                    Em  
We're either riders, or fools behind the reigns
             C
I've spent 

Chorus II

Bridge:
C
            G         D
It keeps on falling
             Em       C
It keeps on falling
            G         D
It keeps on falling
            Em     
It keeps on falling
        C              G               D    
And the water keeps on falling from my eyes

Chorus II

Ending:

 C        
              G      D     C
It would be a lie to run away