It depends on what map type. For 1v1s, all defaults, here are my thoughts...
On inland and hills (narrows maybe, I have never played narrows) the ability to boom is not that great/predictable, so rushing civs that can advance faster are the best. That would mean Shang, Yassy, and Rome. On hills you could add Sumer and Minoa because of their superb farms and Iron Age militaries. But Shang/Yassy/Rome will usually get the first decisive punches thrown.
Conti is also pretty much a rush map, but you might be able to get away with a Mino/Hitt booming which would have a better military.
Medit is the ultimate boom map. The bigger it gets, the better Minoan/Hittite gets. Mino of course has the great economy and the super composites and dominating navy, while Hittite has the outrageous boats and the most dominating bronze (second only to mace) and iron in the game once it gets going. Rome is also a good boomer, but I can't see them outbooming Minoan. Shang can boom well too, and they can rush faster than anyone else. Whether their rushes comes fast enough anymore, I'm not quite sure.
With the departure of Phoenician, islands is wide open. Yamato can dominate in tool. Their ship bonus is powerful in that it doesn't require ships to be massed like Hittite, or many docks like Minoan. However, Yamato must watch out for Egypt which will just ignore the boats, go bronze, and convert them all. Ditto (sometimes) for the Minoan. Hittite's boats can hit Egy's priests with no problem and Egy dies to them horribly. Hittite's FG/war galley also dominates Iron. Minoa's cheap boats are awesome and if they manage to land you with composites--uh oh.
So here is my civs ranking overall:
1. Shang--they are still the best because of their versatility. They can rush, they can boom, and they can do it faster than anyone else. In bronze when the villager booms start they can gain the upper hand, saving 10 food on each villager they make. An early Iron of FGs against boats, Cats against compies/ST/CA, and HA against everything else is very powerful. With the patch their window of opportunity is much smaller though. If Shang lets their enemy build up too long they could be in trouble.
2. Hittite--They're slow but they have the best army and navy once it gets going.
3. Minoan--They can outboom everyone else and then blast away with great composites. Did I forget to mention their navy? Doesn't do so well against Hittite though, so I put them a notch below Hitts. On non-boom maps they aren't as powerful but they extra output farms are cool.
4. Roman--Faster then you'd think and a very nasty tool rush. Cheaper buildings=better boom as well. In bronze though, they must rely on towers and ST on defense, which sometimes just doesn't cut it. If they can Iron, scythes+helos+hcats and legions/cent if you feel like getting them are very hard to stop.
5. Egypt--I don't know where to put them. They're slow like Hittite, but a great military too. After nobility their CAs and chariots rock. In Iron the scythe waves never seem to end, and before you know it your gold troops turn around proclaiming themselves Egyptian.
6. Assyrian--Fast villies guarantees you forward building and a very nice rush, but if you don't do major damage with your rush you could be in trouble. Unlike Yamato they have a decent if not dominating Iron.
7. Yamato--same as Assy, except your most dominant timeframe is a little earlier with the scout rush. Follow your scouts with cav and then composites and you might have something going. Buff boats really rock. If your opponent Irons and you can't kill him quick with HA hoardes you might as well resign.
8. Macedonian--Composites and cheap ST kill everything in late bronze. Make hoppers if you feel like having fun. Just hope that you don't get hit early because your non-wheeled villies are in trouble if they do. Early Iron rocks but once the heavy siege starts rolling in the trouble starts.
9. Sumerian--They can kill everyone except Hittite in Iron. On smaller maps a villager rush is very annoying and effective if you can pull it off.
10. Bab--Super veratility, but they're too darn slow. In Iron they can hang with Rome and beat everyone else except Hitt/Sumer. HCats/HA/Tower/Priest in Iron=lots of dead gold civs.
11. Phoe--Hey they still cut slightly faster. Versatile like Bab, except they get a much crappier Iron.
12. Choson--Hey, if your facing a slow gold civ you can try making tons of priests and towers. Iron jumping is fun but usually doesn't work.
13. Persian--If there's a lot of stuff to hunt then they can vault into the top 5 with a very fast and nasty rush. If there's not like on most maps then they really suck. No need to talk about their horrible econ which can't make its cool Iron units. Legions would be a good idea if the scythe wasn't invented.
14. Carthage--Might be fun to slingshot as their early Iron of HAs and FGs is awesome. Surviving to early Iron is another matter.
Greek and Palmyran are so bad in 1v1s, I don't know where to put them:
15. Palmyran--You gather faster and spend it back (and then some) on villagers. You're guaranteed vill lag. In other words any well executed tool rush will wipe you off the map. They also have have no dominant timeframe militarily. They have all bronze units but Shang, Bab, and Phoe do it better. They have HCats but without engineering. And finally their fat villagers are not better than anyone else's in Iron as they miss craftsmanship and coinage. Their farms are cool but run out very fast.
16. Greek--they don't even get the friggin improved bowman! They're a slow civ with no military until mid-Iron when they can mass up siege. That hardly ever happens. If they can make it to mid-Iron healthy than they can beat everyone except Hitt, Sumer, Bab, and Rome. But that will never ever happen against competent players.