Well, I'm not just curious about UUs. Essentially, some units do not specifically state that they receive a bonus against another type, so it is unclear what they are used for to counter. For example, what does one use to counter a SoD of macemen? Sure, you throw a seige at it to weaken it, but what is the preferred unit. I think there really needs to be a list for this somewhere. It's amazing that there isn't one.
Warrior - anything
Archer - UU's, otherwise just units with CR
Spear - axe, xbow
Axe - Er...axe. Horse archers will win this once you have them, same with xbows. Chariot (Thanks mirth! Total brain fart here on my part!)
Sword - Axe. xbow
Horse Archer - Spear, elephant
Catapult - Horse archer or anything that manages to attack it
Elephant - Spear.
Longbow - Siege basically
Xbow - Horse archer/knight
Pike - axe/mace/xbow
Treb - knight
Mace - Xbow
Knight - Pike
Musket - Knight (muskets beat everything other than knights from earlier eras)
Cuirasser - pike, rifle
Cannon - cuirasser/cavalry
Grenadier - Cuirasser/cavalry
Cavalry - rifle
Rifle - grenadier
Infantry - Machine gun (only defends), will dominate anything earlier. This makes infantry one of the most powerful military break points in the game. Tanks beat them finally.
From here it gets a bit more complicated with air and naval combat. Marines aren't great on land (they lose with equal promos to infantry, and don't particularly beat anything else) but can dominate amphibious strikes. Fighters are only countered by things with interceptions - air superiority owns pretty hard. Nukes before SDI will of course beat any stack of anything easily, etc etc. Counters break down a lot here - but do mind the obvious ones like gunships vs tanks and infantry vs marines/paratrooper, and remember SAMs and now MG's both have interception chances on land (at sea, destroyers can intercept).
Also note that although the suggested counter units often work best, any unit with a higher strength that isn't countered will win too. This means, for example, that war elephants will easily beat longbows after collateral damage, but this is not the conventional approach oftentimes. If it gives you the best odds, use it though.