Where I live it is illegal to attach anything to the widescreen except on a small corner on the lower left or upper right side of the glass, so dash cams are technically not permitted. But these laws were put on the books before people started attaching GPS units, toll responders, radar detectors, dash cams and other gadgets to their windshields and so the police don't seem to enforce the law at all. They also seem to have a lot of stuff attached the glass on their own police cruisers, so maybe that has something to do with it.
I remember when I first got a dash cam I was very worried I would get pulled over by the cops because of it as I was familiar with the law but nothing ever happened. Then one day I was at the local supermarket when I noticed that I was parked next to the truck belonging to the recently retired police chief of our local town (city) and he had a dash cam mounted on his windshield and then I knew I didn't have to worry about this anymore.
Anyway, I'm pretty sure that a big piece of cardboard attached to your windshield would still probably catch the attention of law enforcement. The original laws were partly about attaching signage to your windshield as well as electronics and lights so I think a piece of cardboard with suction cups could be a problem.
