Step 10: Split Your Artwork Into Three Pieces Along Cutlines
There are hundreds of ways to do this in each program... the following examples are just one of many.
Put the cutlines and the Logo/Artwork on the same layer, with the cutlines on top. Select the cutlines and then Object>Path>Divide Objects Below. In addition to cutting along the "seams", this will also cut each panel into two parts (meaning we will want to then rejoin them). Rejoin them by selecting the two parts and Alt+Clicking on Pathfinder's "Add to Shape Area".
Ctrl+click on the layer containing the outline of a specific box to select the area of that box. Then select the layer containing the logo/artwork. Copy, and paste to new layer. Repeat for the other two boxes.
Again... I'm sure there are many better ways to do this. These are suggestions only. I was trying to keep it simple so as to make it accessible to the most people.