Alright, so most of the versions for this song were fairly accurate with the 
general chord progression of this song, however, i believe that there were some 
major changes that could be made to improve the overall quality of the song.
 
Major change- The Em in most of the versions was simply not doing the song 
justice, by making it an Edim7 in most places it adds greatly to the jazzy tune of the song.

Here is how i play the chords...
    Bm     Edim7       F#7       Em        D       G7
e|--7---|----X-----|----9----|----7----|---5---|---10------------------------
B|--7---|----8-----|----11---|----8----|---7---|---12------------------------
G|--7---|----9-----|----9----|----9----|---7---|---10------------------------
D|--9---|----8-----|----11---|----9----|---7---|---12------------------------
A|--9---|----X-----|----9----|----7----|---5---|---10------------------------
E|--7---|----9-----|---------|---------|-------|-----------------------------

- Keeping the chords relatively close to each other makes it easier to switch 
between them and makes the song sound smoother and sexier.

* The G7 is in the original song, however it sounds totally fine without it and 
may simplify playing it.

     Bm  Edim7 Bm     Edim7      Bm   Edim7 Bm
Summertime,   and the livin' is easy
         Em                        F#7  G7* F#7
Fish are jumpin' and the cotton is high
             Bm  Edim7 Bm       Edim7        Bm    Edim7 Bm
Your daddy's rich,    and your momma's good lookin'
   D                F#7        Bm  Edim7 Bm
So hush little baby, don't you cry

Bm                  Edim7 Bm          Edim7      Bm  Edim7  Bm
One of these mornings,   you're gonna rise up singing
            Em                                        F#7  G7* F#7
Then you'll spread your wings, and you'll take to the sky
              Bm    Edim7 Bm       Edim7        Bm    Edim7 Bm
But till that morning,  there's a nothin' can harm you
     D                F#7       Bm  Edim7 Bm
With daddy and mammy, standing by

Hope you enjoy!