BomberPlane474: Like Samez said, multiple cores can't help run Civ 3 which is the biggest detriment to fast play. The best I can offer is to make absolutely sure that nothing is running in the background before you load Civ 3. I always check to make sure stuff like Steam isn't quietly sucking away horsepower when it's needed most.
Samez: You really turned the game around there. I fixed the Neutral player adjective. I am very pleased that the Soviets use their artillery so well in this set up. I also like that for escorts they often use a combination of ground units and air defense, which makes them even harder to crack open.
NATO by design actually has a slight edge in the race to military alliances. That is if a major NATO player maximizes research and goes straight to Diplomatic Alliance. Depending on who I am playing, sometimes I will take this route even though it means not having any improved units.
The free barracks and POMCUS sites brought a lot of agony in development. The alternative is having the US magically airlift their units to cities without airports and then getting them stuck there. This happened so much in the older version that they were practically useless. The sites do go inactive with the International Trade tech, which is the one that opens the Atlantic. It's a trade off either way. I'd be open to ideas too; this is the best that I was able to fix this issue.
The Soviet Navy can be a lot of fun. Like Lionic pointed out, enemy air units will crush you if you are not meticulously careful in how you deploy your ships. The Oscars are you best asset, since their invisibility gives them the chance to sneak under air cover and do great damage. As the Soviets, I like to take out the carriers first, then the amphibs and transports. I expect to lose at least 2/3 of my ships. The Kirovs are powerful but again must stay out of aircraft range. You guys probably noticed that the AI goes completely ape when it gets transports in range and will attack with everything at hand. For the Soviet Naval Infantry, I like to use them in local operations in Norway and Denmark until the naval and air situation shifts. Then you can go into Scotland and Iceland and do all that crazy stuff. I have yet to try to invade North America...I think the AI would go nuts and it would be hard to hold territory. In the newer version of the game, naval powers will have the late game option to make new marines and amphibious ships so that you can get creative if you like.
I agree with everyone that keeping naval bombardment is best. Was just thinking out loud there.