How to get very smooth, flat cooked potato pieces
For Potato Jenga you are going to want to use a French Fry slicer like this one from Amazon. The larger slicing grate will produce potato of about the right square size. You will want to exclude the edges and end pieces.

Some additional tips: several hours to a day before you are going to serve them slice and blanch the potatoes in sweet water (boil water, add sugar, and potatoes boil for about 5 minutes, remove potatoes to and ice bath and store in the fridge.) Just before frying, dry the potatoes on paper towels to remove surface moisture. Once they are golden brown remove the fries to bowl and toss them with a light coat of olive oil.
[Edit added on 12/8/11]

Crispy Pototoey Goodness: Success
Jenga: FAIL

I love questions here that cause me to experiment with food. So after answering this question I decided to try this myself. Well, the potatoes came out nice and crispy and tasty, but I am afraid that @shujaa is correct, no matter how you cook them, you aren't going to get a good game of Jenga out of them...

You can get a good &quot;Jenga Like Tower&quot; this way, but you wont be able to play with your food...
Might I suggest carrot sticks sliced to the right size?