I used to never, ever, ever use Republic in my games. However, in C3C, I have actually now started to use Monarchy -- my first time ever.
In one game I made TWO wasted anarchy periods (I cheated -- when I tried to go Republic I reloaded and tried Monarchy, then reloaded and just did Despotism again -- I was embrioled in war and even Monarchy had too high of upkeep costs). The next time, after I was in Monarchy, I tried Republic but had to go back to Monarchy (I don't do more than 5 turns autosave, so I just basically went to anarchy to go back to the previous government -- the revolution failed!)
I have not tried communism yet (650 Mhz PC does not allow for particularly large games).
It looks like Fuedalism could be nice for a very, very tight city build.