The best chance the Roman army would have had against zombies is thus the post-Marian army, which was able to campaign for long periods of time and which was rather tactically flexible, allowing for maneuvers and quick, orderly formation changes. By that time in history, Rome was in full control of Italy, and would have been able to deploy a legion from anywhere to counter an incoming zombie horde with relative ease. Still, the zombies would have posed a serious threat to the Roman army. Warfare in this time period relied on hand-to-hand combat, meaning that a Roman soldier would have been much more threatened by the zombies than a modern soldier armed with a gun. However, the Roman armies were trained to fight in tight formations thus inclined to counter an uncoordinated zombie horde as a single, coherent force. Using such tactics and formations, it would have been nearly impossible to sweep the countryside, rout out all zombies, and eliminate any stragglers. Notably, in battles against tribes and peoples living in the Roman hills which fought the Roman army in an uncoordinated manner (something like how zombie hordes would have attacked), the Romans found themselves taking serious losses. So, most likely, Rome would have been overrun.
A few things come to mind here:
1) Zombie General ? Ok, perhaps not ! How about Zombie Chieftain ?
2) Human players aspiring to... zombie leadership and eventual glory ? Hmmm, intriguing, isn't it ?
3) Waaait a minute ! Something fits like a glove here ! The Programmed Opponent (i.e., Artificial Intelligence) would be a natural zombie chieftain, right ? I mean, lack of coordination, hordes of sacrificial units and so on...
4) Zombie briefings ? Where are the learned linguists now that we really need them ?
