Written by Irving Berlin, as performed by Taco Ockerse. This is probably a pretty
simplified version. For the most part, it seems to fit Fred Astaire's recording as well
(specifically, the recording found on Astaire's Dancing Cheek to Cheek collection). 
Though, for adapting this to any of the other numerous versions of Cheek to Cheek, bear in 
mind that
Taco's rendition is performed in C, and modulates to D.

C (x32010@1)    F (133211@1)     C (x32010@1) FHeaven, (133211@1)I'm in heaven
       C (x32010@1)         Em (022000@1)    Am (x02210@1)   Dm (xx0231@1)    E-A
And my heart beats so that I can hardly speak
      Dm (xx0231@1)             G7 (320001@1)          E-A
And I seem to find the happiness I seek
           Dm (xx0231@1)         G (320003@1)           CWhen (x32010@1)we're out together dancing cheek to cheek

Heaven (cheek to cheek), I'm in heaven (cheek to cheek)
And my heart beats so that I can hardly speak
And I seem to find the happiness I seek
When we're out together dancing cheek to cheek (cheek to cheek)

G7 (320001@1)                C (x32010@1)              G7 (320001@1)              COh, (x32010@1)I love to climb the mountain, and to reach the highest peak
       G7 (320001@1)              C (x32010@1)             F (133211@1)     G7 (320001@1)    CBut (x32010@1)it doesn't thrill me half as much as dancing cheek to cheek
      G7 (320001@1)         C (x32010@1)           G7 (320001@1)       COh, (x32010@1)I love to go a-fishin', in a river or a creek
      G7 (320001@1)           C (x32010@1)             F (133211@1)     G7 (320001@1)CBut (x32010@1)I don't enjoy it half as much as dancing cheek to cheek

Cm (x3101x@1)                     AbDance (xx1114@1)with me, I want my arms about you
    Bb (x1333x@1)                 C (x32010@1)      Dm (xx0231@1) G7The (320001@1)charm about you will carry me through to

C (x32010@1)  F (133211@1)                        C (x32010@1)     FHeaven (133211@1)(cheek to cheek), I'm in heaven (cheek to cheek)
       C (x32010@1)         Em (022000@1)    Am (x02210@1)   Dm (xx0231@1)    E-A
And my heart beats so that I can hardly speak
      Dm (xx0231@1)             G7 (320001@1)          E-A
And I seem to find the happiness I seek
           Dm (xx0231@1)         G (320003@1)           CWhen (x32010@1)we're out together dancing cheek to cheek (cheek to cheek)

[The song modulates here, moving up to D. I CANNOT figure out the chords to this section
-- if anyone could help me out by putting them in the comments it would be greatly
appreciated. What follows is my best guess]

CEm   C (x32010@1)Dm (xx0231@1)Em (022000@1)Am (x02210@1)DmG  E (022100@1)A (x02220@1)Dm (xx0231@1)      Dm/C (030231@1)            BbDance (x1333x@1)with me, I want my arms about you
    C (x32010@1)                  D (xx0232@1)      G (320003@1) AThe (x02220@1)charm about you will carry me through to

D (xx0232@1)    G (320003@1)     D (xx0232@1) GHeaven, (320003@1)I'm in heaven
       D (xx0232@1)         F#m (244222@1)    Bm (x24432@1)   Em (022000@1)  F#-B
And my heart beats so that I can hardly speak
      Em (022000@1)             A7 (x02020@1)         F#-B
And I seem to find the happiness I seek
           Em (022000@1)         A(7 (x02220@1)fr.)             DWhen (xx0232@1)we're out together dancing cheek to cheek

A7 (x02020@1)           D (xx0232@1)               A7 (x02020@1)          DI (xx0232@1)don't wanna climb no mountain, I don't wanna swim no creek
       A7 (x02020@1)      D (xx0232@1)                  G (320003@1)   A7 (x02020@1)DAll (xx0232@1)I ever wanna do with you honey is dance cheek to cheek

Dm (xx0231@1)      Dm/C (030231@1)            Bb (x1333x@1)    A (x02220@1) Ab (xx1114@1)        DmDance (xx0231@1)with me, I want you to dance          cheek to cheek!

Taco's official site:
http://www.taco.tc/