It seems to be totally random, with just a couple of contributing factors. Sometimes it doesn't happen for ages, and sometimes it'll happen right away. Usually you only get one piece, but I had a really weird game where I got nothing for a whole era then got EIGHT pieces at once!
The whole of civ3 seems to be based on randominity rather than anything else. Take culture, for instance. In one of my games I had managed to get pretty much all of the cultural wonders in the game. Every civ was "in awe of" my culture. And yet, I only took over one city at a time, every 20 or so turns. On another game, I had a pretty low culture (in fact I think I was in the Medieval age, so I hadn't even had the chance to build most of the cultural improvements and wonders)... but on that game, I took over THREE cities culturally in the same turn (and it was a Small map, too!). That was really weird.