I have made the experience that even with such a dense object there might still be some holes that you can look through. To prevent this from happening i put a smaller cube in the middle (pic 22) and attached it to the rest (select the "manycubes-cube" and in the editable mesh-rollouts click "attach" and then the big cube inside. Click attach again to leave the attach-mode). It is now part of that object. You'll now have a lot of redundant geometry in the middle, you can delete that if you want. Selecting the inside without selecting the outside is a little tricky, you can either select all and then deselect the outer parts or use the volume select modifyer. You don't really have to delete the inner part, it's just very usefull if your computer has troubles coping with all that geometry.
Now let's start the mapping: First we need to assign the images that were created in illustrator to the materials. Click on the first submaterial, click on the grey box next to the diffuse color and chose "bitmap". You will then be asked to select the bitmap from a destination. Locate the first image-file and click OK. Go back to the material settings and change self-illumination to 100. We don't want any shadows to destroy the illusion of looking at a flat planar image, self illumination will prevent any shades and shadows. Do the same for all submaterials, assigning different images to each one of them. If you now clicked on the "show map in viewport" toggle you would see that the image isn't mapped properly: each little polygon shows the entire image, so you see hundreds of little images instead of one big one over the entire surface. We will fix that by selecting material ids and using the uvw map-modifier: with the mesh selected go to polygon-subobject mode and in the surface-properties rollout click on "select by id" to select all faces of one colour (bottom of pic 23). Chose id number 1. Without leaving subobject mode add a uvw-map modifier. By leaving subobject mode active you only map the active selection, leaving all other polygons untouched. You can tell that the modifier is set to a subselection by the grey box that appears to the right of the map modifier in the modifier stack. Leave the mapping method set to planar. Make sure that the alignment is set correctly (at the bottom of the modifier you can chose x,y or z as shown at the bottom of pic 24), the gizmo (an orange square) should be parallel to the selected faces. Click on "Fit" to make the dimensions of the gizmo fit the borders of the mesh. First face id is mapped correctly now. Pic 25 shows what it should look like now: the yellow square (active mapping gizmo) surrounds the selected polygons which are shown red.
To check the result go to the first submaterial and check activate the "show map in viewport"-toggle (pic 26). In a shaded viewport the result should look similar to pic 27. You have to make sure that you are looking from the correct view, if you don't see your map then hit t,b,l,r,f,k (shortcuts for top,bottom,left,right,front,back) to find the viewport you need. In this example my map was mirrored (pic 27), which of course was not what i wanted. That's because it was mapped from the wrong direction (mapping gizmo turned 180 degrees). If this happens to you, don't worry. In the material editor go to the map-settings and adjust the uvw rotation. Here i had to rotate it 180 degrees around the v-axis. That did the job. The other way of doing it would be to rotate the gizmo - whatever method you prefer.
Do the same for all material ids: add a "mesh select"- or an "edit mesh"-modifier and in polygon subobject mode click select by id, select the next id number, add a uvw-map modifier and so on. In the end you should have 6 UVW Mapping modifiers assigned to six different material id selections.
If you render one of the non-perspective viewports (front, top, left ...) then you should now see one of your illustrator images as if it was a plane 2D-image.