Some Jenga purists may feel like this is the John Henry vs. the Steam Driver reincarnate, so proceed with caution.
Here is the break down on the parts needed for this game changer. The creators of the Jenga Gun have the instructions available for sale as well.
(via Woodgears)
The pistol works by hurling a wooden captive bolt forward inside the pistol, powered by rubber bands. The bolt has a pin on the front, which protrudes out of the muzzle of the gun by about 2 cm at most. This is enough to transfer the momentum of the bolt to the Jenga block in front of the gun, not unlike how the game of croquet works. 
I actually designed the pistol so that the rubber band pulls the bolt back again from its front most position after impact. The main reason for this is that if I need to make a shot where I shoot out the sole supporting member of the Jenga stack, that the bolt withdraws a little bit after impact to get out of the way of the Jenga stack coming down to fill the gap. With a bit of practice, this maneuver turns out pretty easy to pull off, without collapsing the stack.
Other than the rubber band and pulley, the bolt and trigger are the only moving parts of the gun. The bolt has this little keyhole-like cavity cut into it to hook the rubber band into. I was perhaps being exceedingly clever in doing it this way, but it's the only way that I could think of that would allow easy replacement of the rubber bands, and allow the rubber bands to pull the bolt forward, and then back a bit at the end of its stroke. 
The trigger piece is two pieces of wood glued together. It has a hook for holding the ends of the rubber bands. This also serves to pull the trigger up against the bolt from the rubber band tension. 
When the bolt is pulled back, it automatically hooks onto the trigger. I actually carved a very slight hook shaped incline into the corner of the bolt where the hook catches onto it, to make the gun less likely to trigger accidentally. Cocking the gun is a simple matter of pulling the block all the way back.
The other little tricky bit is the pulley. There's a little wooden block glued to the rails (facing the camera in the shot at left) to keep the rubber band from moving too far sideways on the roller. Without this block, the rubber bands were prone to get stuck between the roller and the piece of wood holding it.I varnished all the moving parts, and put a few drops of oil on the pieces the bolt slides against to let the bolt move more freely. I originally designed the gun to use just one rubber band, but with two, it shoots harder, which is more fun. It's enough to get the bolt going to about 12 meters per second, and accelerate the impacted Jenga block to about 6 meters per second.
