SHORT SWORDSMEN/BROAD SWORDSMEN: Don't see much of these but Camels are decent against them. Where possible, I'll use Cavs instead...
CAVALRY: Obviously countered by Camels.
CAMELS: Countered by Persian Camels and supported by whatever units are handy (Slingers and leftover Clubbers/Axers most likely.)
CHARIOTS: Camels and Slingers again.
ALL ARCHERY RANGE UNITS: Once more the Camel/Slinger combo is what I go with here. In the case of Chariot Archers, Camels close the gap and give the Slingers a chance to move into position. If the Chariots run, the Camels follow. If they stay and fight, the Slingers tear into them. Improved and Composites are harder hitters but less mobile.
HOPLITES: Ugh. The ONLY effective counter here (short of walls and towers) is Priests. A half-dozen Hoplites can kill six Cavs, six Slingers, and two Sentry towers (sometimes there'll be one tower left standing.) That's just for demonstration as I rarely build towers anyway. Better to spend the Gold on Priests and gain a couple of these guys since you can't build them yourself. Or you could double-wall and pick them off with Slingers...
STONE THROWERS: Cavs are best here with hit points to survive supporting archer fire and deal more damage when they reach the target.
PRIESTS: For the same reason, Cavs work here. Also, if they're converted, your Camels can deal with them better than vice versa. As for Priests behind walls, maybe it's time to reconsider that Siege Workshop...
TOWERS: Slingers (with Camels or Cavs going after the builders.)
WAR GALLEYS: Your own ships (if you have them) are outranged at this point and your woodcutting is falling behind, so what do you do here? Priests? Abandon the coast? Seems like when I see Galleys it's usually in bunches...
SLINGERS: Not a Bronze unit but still used there. Do Cavs get their charging bonus against Slingers? If not, they're still the best choice, unless you've got a horde of swordsmen (why?) or leftover axers (and Bronze shield.)
VILLAGERS: Whatever's handy. Villager killing is always a high priority.
Like I said, if anyone has any suggestions for better matches, I'd like to hear them.
Also, if I had my way, Persia'd get the Wheel and Chariot (and Scythes) to make up for their Range deficiency. They could also use Siegecraft to make their Slingers fully upgradeable. That would ALMOST give them a decent ranged unit. And throw in Coinage for the late game (all of Persia's post-Tool land units cost Gold.)