In the new expansion pack for Civilization III, Conquests, the Rise of Rome scenario centers on a conflict between the Carthaginians and Romans but there are some decent opportunities for the Persians or Macedonians to find their own paths to supremacy -- more so if the Romans and Carthaginians hit a stalemate.
The Fall of Rome campaign is a bit different. There are now two halves of the Roman Empire and a large number of barbarians ready to tear through either of them. As a barbarian leader (you can't play as either Roman civ), you'll have to attack Roman cities. Once eight cities from any civ are destroyed, that civ is removed from the game. There was a variant of this kind of play in the previous expansion and it makes for some thrilling moments as one side goes on an all-out attack against a well-defended enemy. Since the cost for the Romans to reach the Barbarian side of the tech tree is outrageously high, each side in this struggle will have to adopt a unique approach to their problems.