I've been seeing a similar situation to most it would seem.
As my first city was building a spearman I sent my worker south acting as an explorer. I discovered the Aztecs and began working a road backwards as I knew I would need to get a city near there ASAP. My spearman wandered about defending my capital and my worker from barbs and did quite well - made it up to elite.
Once my settler was ready, I sent it south right away to be a buffer and staging area against the Aztecs. I was later able to establish a third and fourth city with usable protection (spearmen only), and I got The Wheel from huts and traded the Aztecs Masonry for something as well. Anyway, the time came to build barracks as I was studying iron working and immortals were about to take the field.
I was able to contact the Babylonians as they sent a unit up through Aztec lands. It was actually a bit scary as they were red in color and I'm used to THAT color being the barbs. Anyway, they had a fair amount of tech, and I was able to buy it rather cheaply. Nice.
So now began the art of rush building.
I don't think that too many of my cities ever got to size three. I rush built EVERYTHING for the most part. Crack the whip, here's your barracks. Crack the whip, here's your immortal. Crack - Temple. Despotism is actually a GREAT early wartime government and I'm still in one in the 1200's even though I'm through the whole first tech tree and have discovered both Monarchy and Republic.
Once I felt I had sufficient numbers of immortals, the carnage began. A horde of my veteran immortals took the Aztec capital along with the Oracle first thing. From there, it was just a matter of marching. There were some bouts of resistence and counter-attacks, but for the most part a steady stream of immortals rush built from my barrack-filled cities was too much. In the mean time, the Babylonians and the Zulus made sure they sold me all of their techs so I did not fall behind.
In the cities that I took over, I usually left a single unit - preferable one that was wounded - in the city while one or two units waited outside (great strat. Thunderfall). The new cities build a spearman and then barracks and then immortals. Occaisionally, I'd throw a temple in there as well. I did have the Oracle now after all. In fact, is a city with sufficient size I would rush build whatever I could as quickly as I could (after waiting a turn or so for a shiled or two to be added first). This worked well because it used up the foreign laborers. A city with 6 Aztec laborers could rush a temple, barracks, spearman and then an immortal over 7-8 turns and then only have a single Aztec laborer left. This meant that my Persian laborers equalled or exceeded the Aztec ones very quickly.
In any case, I cleared the continent by about 1000 AD, but all three of my former neighbors were able to settle on distant islands. At about 1200 AD I finally did away with the Babylonians and the Aztecs are nearing extinction. The Zulus have a single city left, but they are on an island populated with Iriquois and I didn't want to have to supply that front yet.
I've had skirmishes with the Americans, and I sent a force over to their continent to try to take some cities. However, they have discovered Feudalism and now have veteran pikemen who my immortals cannot seem to defeat very well.
Now, all is settled down and my immortals have played their part, the rush building of improvements is continuing in earnest. I will rush build with citizens for a short while longer and then switch to Republic in an effort to take a tech lead and get to Democracy later once it's discovered. I can also let my cities grow at that point
. Perhaps I'll try the Americans again once I have cavalry. I have not built a single wonder, but I captured the Oracle, Pyramids, Great Library and the Lighthouse. The Americans have the others and will likely have Sun Tzu's soon as well. Once I get a lead in tech, I'll be able to build some on my own - and likely take some from Abe as well
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I just finished GOTM1, my first, and was a little dissapointed the way it ended. I was cruisin around the map, mopping up any outpost colonies the AI had founded. After finishing I turned my knights towards the three remaining American cities on their main continent. Before I could conquer them all the AI gave me a Dominance Victory. I wanted all there cities.
). First city builds: barracks, spearman, settler. By this time or very shortly thereafter iron working should have been discovered and if you are lucky (which we are on this map) there is a source of iron nearby so your second city can form near it. Build a road to the iron, pump one more spearman from your first city and build a barracks in your second city then switch to straight immortals. In my game its 1000 BC and I've wiped out the aztecs and taken three cities from the babylonians while only losing one immortal. Naturally i traded for their techs before attacking them 
. I guess you could call my strategy immortal rushing. The disadvantage is the lack of early scouting and i was probably lucky that i sent my first immortal south instead of north or the aztecs might have been a little tougher to take out. Another disadvantage is that my golden age is already over 
However i now have 3 elite immortals and therefore a good chance to get a leader sometime soon. Still not sure if im better off using it to rush a forbidden palace (corruption is brutal in civ 3 
Welcome to CivFanatics, Javier. 
Oh well, I'll try to do better in my second game. 
