For me, there are just too many to list. The only problem is that you (or at least I) don't get to enjoy them much because the civ that they are associated with tend to not be that great. But if I had to choose my top five, they would have to be these:
1) Immortals: they can kill just about anything during the ancient times, and still kick ass in the early middle ages. I even sometimes just keep around some elite ones because their swords are pretty awesome.
2) Panzer: Got the edge over normal tanks w/ their movement rating of 3. As much as this seems like only a small advantage, it truly can make the difference. If they're next to a city, they can attack it twice in one turn, b/c it takes two movement points to attack and return to your previous square if there is another unit defending the square you attacked. The only other unit capable of this is M.A. which is some techs later.
3) Infantry: These are the most powerful defenders up until mech infantry or M.A. And in situations where all of your offfensive units have been killed, they can still get the job done with their attack rating of 6. Like Gfeier, I too like to build maginot lines if I have the time with an industrious civ. Even in real life they played a major role in the 21st century.
4) Marines: You hear about them in the news, you watch what they have to endure in basic training on the Discovery Channel, and the only unit that can launch an amphibious assult in the original Civ III. Sure, they die a lot, but what can you do? If Infrogames made them as good as they are in real life, there would be no other unit to match their greatness.
5) Legionary: They've got the defensive rating of a pikeman, and the offensive rating of a swordsman. They can be quite helpful, since they play two roles during the ancient and middle ages. Too bad they have to work with one of the weaker civs and have no upgrade path in the original Civ III.
Other units that deserve recognition are tanks, artillery, cavalry, M.A., and as weird as it might sound, helicopters. They might seem like a waste of 1 gpt, but if you've got a ton of marines and infantry that aren't doing diddley squat, build helicopters en masse to transport foot units with only one movement point up to six squares away from any of your cities into enemy territory. Pretty useful for landing units on the first turn of attack on the enemy's resources, at the front gates of their border cities, and creating an irresistable diversion to the simple minded AI.