With my preliminary thoughts in mind, I chose to be Saladin of Arabia (philosophical/spiritual) for this game. Map size was standard, game speed normal, and because I really wanted to get some experience with diplomacy, I chose to play on a Pangea map. With my three cities only, this would make the game even more of a challenge, as all AIs will have contact with each other early, which quickens the tech pace considerably. Additionally, being allowed only three cities would mean I would never be able to build the Globe Theatre, Oxford University, Wall Street, or more than one cathedral.
My settler started near a tribal village, and founding Mecca automatically explored this village, giving me maps of the surroundings. It was a very nice starting position! I had access to fresh water because of a small river; Mecca was a coastal city, but thanks to an unusual land layout had only access to four coastal tiles which meant not much productivity would be lost; and it had deer, crabs, horses, and wheat in range! Also, a source of marble could be seen to the east, which will be a top priority to get, as it will speed up the building of crucial wonders.
I sent out my warrior exploring, and started another warrior in Mecca. I set the governor to emphasize food and commerce in the beginning, as I wanted to research some key techs quickly! I wanted to make sure Mecca would become a holy city, so I set out to research Polytheism first. With the short river and without floodplains or improvements though, Mecca was growing only very slowly and had not much commerce, making research slow as well.
My warrior explored several tribes and got some lucky breaks at first: The first two both gave me scouts, which relieved Mecca of producing more units for scouting (which is very important on a Pangea map to make contacts)! The third village was hostile, though.
Four angry barbarian warriors came out of the hostile village! What I found hilarious is that on the same turn, a forest grew just where the village had been, so the angry barbarians must have been quite shocked to see that all of a sudden, I had a nice defense bonus...
That helped me only to kill the first attacking barbarian though, and my warrior succumbed to the second already. Ah well, good thing I had gained two scouts!
Mecca was slow growing indeed, so when it finally hit size 2 in 3560BC, I started to build a worker to improve the wheat and the deer tiles. The slow growth and the lack of commerce also made my research efforts towards Polytheism quite slow, and I worried some other spiritual AI might beat me to the tech (and found Hinduism)! What made me even more nervous was the fact that Buddhism wasn't founded. Normally, Buddhism falls first because the tech (Meditation) is cheaper - but no message about Buddhism being founded in a distant land came! Does that mean the AIs have chosen to go for Hinduism first as well?
Luckily that had not been the case; I managed to found Hinduism in Mecca in 3480BC. Buddhism got founded only after some turns later - maybe I was the only spiritual civ in this game? Anyway, Mecca was my first holy city now, and I converted. In addition to the culture from the palace, Mecca had now also the culture from being a holy city, giving it a nice jump start.
A turn later, one of my scouts got another lucky break while exploring a village...
Fishing might not be the jackpot (masonry, for example, would have been better...), but hey, a free tech is a free tech, and who am I to complain? Especially it's better to have a coastal city and get fishing, as opposed to the last game where I was far away from any coast whatsoever and got the fishing tech...
Anyhow, my scouts had made contact with Frederick, Victoria and Louis by now (whoa, two creative civs at my borders!). After Polytheism, I researched Hunting and Animal Husbandry next so that my worker could improve the deer and the horse tiles, then started masonry. After the worker had been built in Mecca, I immediately started a settler. I was confident to grab the marble city site, as no AI had started near that place, but Germany, England and France all were not far away to the south, so I had to hurry to claim a river spot south of Mecca before they would settle there.
I lost both scouts in 3200BC to barbarians, archers already! After completing my settler in Mecca (and sending him not far away to the south), I invested several turns of production into the Parthenon, waiting until I would have researched Archery. But before I could even start to research Archery, I had to research a different tech first. In 2400BC, my second city Medina is founded:
See how close the borders south of Medina are? But now let's take a look at what tech it has been I was delaying crucial worker techs (mining, agriculture, pottery) and military techs (archery) for...
Et voila, my second holy city! Note that I was sure that Medina would become the holy city for Judaism, not Mecca! If you have several cities at the time you found a religion, the game seems to pick one at random - but not one that is a holy city already (except, of course, you have only holy cities).
I did not convert to Judaism though, at least not yet. I adopted Organized Religion next, planning to use the production bonus for building the Parthenon in Mecca later, and Mecca had only Hinduism at the moment. I emphasized food and commerce in Mecca now as well, as I wanted to bring it up to maximum population as fast as possible.
Finally, I was able to research archery (my capital was still undefended...), then agriculture. All my poor worker had been able to do after improving the deer and the horses was connecting Medina and some resources to Mecca with roads... and while he was busy preparing a road to my future third city site, disaster struck: The barbarians decided to found a city there!
Funny, there I was thinking I would be safe from any other civilization settling there because they were too far away, and now that! Normally I would have built more warriors for exploring, and would have sent one of them to stake my claims on the lands I wanted to settle later, to prevent something like this from happening. But in this game, I hadn't had the time! And no way I would have the time now to build a sizeable military force to raze that city to found my own there - I needed to concentrate on culture instead, and soon! So I had to improvise, and decided to found my third city still in that area, but on the coast instead. I could not afford to lose the marble, but the barbarian city had caused me to lose the rice at least. So in 1240BC, my three legendary-to-be cities had been founded.
As you can see, Mecca was building another worker before resuming work on the Parthenon, and Medina was building a Jewish missionary. I did this because in the meantime, both Frederick and Victoria had converted to Judaism! Remember that one key to success for this 3CC cultural victory game was to keep out of wars at all cost, and to achieve this, you need to have good relations to the AIs, especially to your direct neighbours (who can and will become upset about close borders rather sooner than later!). The main reason for good or bad relations is religion: If you have the same religion as the AIs, they love you, otherwise they hate you. That's a bit exaggerated, but religion can make the difference between a -4 and a +4 modifier for relations!
So now that my two closest neighbours have been converted to Judaism, I planned to convert as well. This would serve not only the purpose of befriending the AIs, but would also generate culture in Medina if Judaism would be my state religion. The only drawback was that I would lose the production bonus from Organized Religion, which I felt I needed to complete the Parthenon, hence the missionary.
In 1080BC, Judaism was spread in Mecca, and I converted. To improve relations even further, I also signed open border agreements with Victoria, Frederick, Louis, and Mao Zedong and Washington, whom I had met earlier. Judaism then spread to Munich on its own, and I felt I had a permanent friend now.
Unfortunately, soon thereafter Mao founded and converted to Confuzianism, but there was nothing I could do about it at that time. At least he wasn't a direct neighbour to me!