Here's the rig I set up to attempt to cut straight edges.
As you can see here, it gives enough control to cut paper-thin pieces. Unfortunately, the wire bends towards the end of the stroke and forms a curved cut at the end of the piece.
The NOVA Open crew have a few wire cutter stations that are affixed on a platform and cut in a similar fashion to a scroll saw or band saw. I'm going to try to use that to continue the work, but I need to do it by this weekend, so I may need to make do with my rig.
I'm also trying to track down who makes this vinyl terrain mat. I have exhausted my Google-fu. Does anyone know where I can order more of these?
Update: The fine crowd on the WAMP forum directed me to Zuzzy, not for these terrain mats, but for mats that are even better and cheaper. Yay! Hopefully, my order will arrive by this weekend.