Can't Make the Intersecting Lines Disappear

For a print, I need to convert our logo to one color. The current logo is seven shapes that overlap. Each shape is semi-transparent. This means that when you view the logo, it's made up of lots of colors where the shapes are overlapping each other. Very pretty.

For the one-color print job, what I've done is convert each shape to white. Kinda boring. So I want to set it up so that the borders which intersect are "cut out" (AKA transparent). This way the logo will maintain its unique design with each smaller overlapping area being preserved. It's all in white though. Does this make sense?

There are no borders on the shapes. They're laid out precisely with no overlap, so even if they had borders, deleting the border wouldn't give me the result I seek. I've also tried selecting all seven shapes, grouping them, and then using all the pathfinder tools to achieve the result. All I get is the whole thing disappearing except a small portion. So I then tried only selecting two shapes and then using all the pathfinder options. Nothing. The logo is all on one-layer.

Any other suggestions? This is driving me absolutely batty.