I was watching a program which were comparing various Big Bang theories
against each other. In it, they said one way to falsify Inflation vs.
Cyclic Universe theories is that a Cyclic Universe will *not* produce
gravity waves. Does anybody know why this would be the case? One would
think that whatever mechanism lead to the BB, would always cause a lot
of gravity ripples.

I found some info on it, but it doesn't really explain what leads to
this conclusion:

The one possible test that they point to that could distinguish the inflation and cyclic scenarios is the expected more sensitive measurement in coming years of a possible B-mode polarization signal due to gravity waves in the CMB. They claim that inflation predicts a significant amount of B-mode polarization, whereas the cyclic model doesn’t.

Endless Universe | Not Even Wrong
http://www.math.columbia.edu/~woit/wordpress/?p=563

Yousuf Khan