You don't need a laser cutter to make this project - it just makes it a lot faster. I used the one at TechShop. It's easy to turn your own 3D model (whether you designed it or found it on 123D as I did), scale it to any size you want, and print out the plans for it. 123D Make even provides directions for how to put it all together.
Step 1: Using 123D Make
If you want to play with the model that I made, check it out here: http://www.123dapp.com/3dmk-Make/Giant-Cat/656520
You can see in the second image that the tool bar on the left allows you to do all of these things, including change the direction of the slice and change the form somewhat.
On the right hand side, it shows you what cuts it plans to make based on the size of your model and direction of your slice. After inputting the size of the cardboard I had on hand, it compiled a tidy little file of all the cuts for me which I then downloaded.
The third image shows you the order for putting your pieces together in a neat little slideshow, and highlights which piece is coming up next. Seems unnecessary until you have tons of pieces in front of you that seem to make no sense whatsoever. It happens.
If you don't want to play with the model here, and just want to make it, I provided the .eps files for you to download here!