The corridors down the side are... corridors!
Basically there's a central courtyard with a raised level at each end. Corriders run down each side at both ground level and a higher level. Stairs at each end of the ground floor corridors run up to doorways to the raised platforms. Stairs at each end of the platforms run up to the first-floor corridors.
My original plan was for a series of connected chambers that could be put together to make a bigger network, but I've decided to just make the one for now and see how it goes. I might still attempt the grand vision eventually. Obviously this board will be quite small and would probably only suit small skirmish scenarios, but as I'm still new to the game anyway this might be a good thing. Also, storage space is a major factor for me.
I'm a bit concerned that the finished thing might look too neat and clinical, and more like a building than a dungeon, so I might add a layer of papier mache or something to give texture to the walls. I'll worry about that when I get the pieces glued together.
Thanks for the feedback. I'll keep you updated on progress. If it's a success I'll write up my methodology as an article, if people would be interested.