I've managed to survive quite a long time as the Aztecs. I'm still behind in tech, but i have enough to protect myself. One of my strategies was to expand quickly, so i pretty much have the whole of north america. It makes it easier to protect yourself if civs who hate you don't have cities on your continent. I would feel safe fighting off a few cavalry if they ship them over, even if i only have macemen. Currently only the english have a cities over in New England, and I have a defensive pact with them.
Also, another strategy of mine as the aztecs is that i went straight for the gold and silver resources. I hooked them up first, and my third city was even all the way up in northern california where all that gold is. It makea a big difference. I also finally killed off all those barbarians, and have their cities now too, plus a few more of my own.
Lastly, I'm debating a third strategy to get me into contention. I'm the second last civ right now (with incas being the last). I think I can hold off any invasions for a while now, so i think what i really need is someway to get me back into the tech race, sooo...I'm thinking about attacking the strongest civ in the game.
Before you laugh, I should mention that the strongest civ is Saladin. He has control over all of north africa, down the east african coast, the middle east, and he took over the whole indian empire. So yeah, he's pretty strong. But, he has a major weakness that I can use to my advantage.
Jerusalem and Mecca (or Medina, I forget).
They're the holy cities for the worlds three biggest religions, and they are right next to each other. And they have shrines pre-built. Which means a lot of gold. Which I think is really the only way to make me a contender in this game. So capture two cities, reap an extra approx 40-60 gpt. Use that to buy science buildings and tech if i can. So I'm debating a huge sneak attack. I know they'll be well defended, with units probably better than my own. But still, a few galleons of catapults, and a few more of macemen, against a relatively weakly defended city. Then of course I'll just have to hold them, but thats easier than attacking. Ahh.. the challenge