Currently, after some time playing on regent and monarch Ive managed to get ahead on the next one up difficulty wise... the key? Be a bastard. The computer does "cheat" on regent or higher in Civ3, although the higher from regent you go, and the nature of the your game actions seems to adjust the level of cheap cooperation the computers have with themselves. For example, lets say your taking down the last few cities of a rival on your continent, they have no strategic resources, no infestructure, nothing... but still in yet there knights and calvary come out every few turns... other comps are aiding them. Anything past regent is in the comps favor in this respect, reduced build times, reduced research times, instant mapping, map knowledge, statistical calculations on combat. What does hold true no matter the difficulty in civ3 unlike 1 and 2, is that everyone plays by the same combat rules; use this to your advantage. Find a computer that is weak and pour on the pressure, make peace and extort lump sums and research, then attack him the very next turn, sac his lux resources, take away anything that is being traded, blockade his towns, catapult and arty the towns into the dust, use workers as carrots to draw out defenders, build privateers to draw in there ships close to your cannons. If you can spare it get an army and win a fight, opening the door for the heroic epic, war college, and pentagon; later in the game 3-4 tank armys make very very short work on stacked unit towns. Playing with more civs on the higher dificulty helps you out alot; by reducing the available land to any one AI; and slowing things down some. With this in mind, your map selection, resource availabilty, and civ you choose should all be planned out early so that 20 turns into a game you know how its going to go. Ie. no horses or iron that you can get too by 20 turns, restart the map. Get wonders with heros, or do the palace build, swap out to get production; build with a purpose; and pass up things that wont help you in the long run. Again, anything past regent with civ3 on large or huge maps will require some experience or youll find yourself pinned in and outgunned while an AI on the other side of the world is flying spaceships and building the UN by 1800AD.
My personal favorite things to do:
UN victory set to off (ive won and lost to this... and its way too anticlimatic for my taste)
Capture the pyramids
Hero build the forbidden palace
Hero Sun Tzu's
Universal Sufferage makes being a commie bastard the way to go.
Palace move - swap to heroic epic
Tech straight for monarchy and or great library (Ive never actually gotten the library on my own but it would be cool)
Get iron or horses asap; if not restart.
Play with a militeristic civ. Japan is great with samurai or horses as your fast blitz attacker; Germany runs a close second with the science bump but frequently have low culture, Panzers will golden age you as will sun tzus and any science based wonder combined.
Rip of the computer, trade lux for a tech, then pillage the improvement on the lux; cancelling the trade peacefully.
Capture workers and put em to work or join em to a city and sac them out again for units or a new settler.
Raze cities to get lots of workers, wash rinse repeat.
Catapults early means artillery later with upgrades.
A catapult and 1 ship is better then 2 ships.
A trieme now is a transport later.
Steal from your friends
Steal from your enemies
Newly conquered towns can be auctioned off, and retaken.
Spearmen hold ground, tanks take ground; dont spend ill gotten cache on upgrading units unless you need too.
Settlers should go with an army on any campaign. (2 is good, one to start the town; one to join with the town and commie sac out for an airport)
Captured workers require no upkeep.
Autosave is your only friend.
War to win, at certain points in your research there are noticible technology breaks; use this time to reduce your research and concentrate on warfare of that age. (This is especially good with the japs as they only suffer one turn of revolt in gov switches). Switch off to republic, use money from the 10% science in the lull between new resources and units to buy up units, then switch back when you start to fight and dig in. The extra units will negotiate the missed research for you, and in the end youll have a few more units.
Elite units should never go it alone unless the risk is minimal, once a unit hits elite; save it to fight low risk battles; they will eventually poop out a general.
A general reduces the chance youll get another one while he is still in play.
The intellegence agency quickly becomes the hoover dam
Use AI driven town management; and auto half your workers; the AI is much better at keeping people happy then my warmongering self ever will be.
Extorted money paid to spys is a good way to get rid of the pesky space ship in that capitol.
Did I mention a privateer next to a town with 10 cruise missiles is a good way to even things up? (Comps home in on privateers, if it sits for 10 turns, youll have most the worlds navy at your door.)
This is also a cheap diversion before you land a ton of weakly guarded transports (from those upgrade triemes) on your neighbors door. Dont forget to stack all those upgraded cannons and catapults at the town as well, along with a baby "tactical" nuke. Use cheapo subs to mop up the remains.
Thats about all I can think of... the cool thing is; theres so many things that you can do to get an advantage, and there all rated as "bastard". Of course, be prepared to restart alot; diety is "nearly impossible" so best of luck with it... Ive still yet to win on it; but im getting there.