Besides the basic non unique units:
Babylon (Near East only)
Babylon archer => Babylon composite archer, add 'Vulture' Sumerian axeman
Persia
'Immortal' spearman, Persian medium cavalry, add 'Kardaksha' skirmisher (armored javelineer-archer)
Assyria (Near East only)
'Redu' spearman, battering ram tower, add heavy (4 man) chariot archers
Egypt
Egypt war chariot, change Nubian mercenary to Medjay bowmen, add 'Sherden' swordsman
Nubia
Nubian mercenary skirmisher, Nubian composite archer, add Amazon Chieftainess (elephant rider, after Candace of Meroe

)
Phoenecia (Near East only)
Phoenecian merchant, Phoenecian bireme (allow 1 unit capacity), add Aramaean Camel Rider (can defend, conducts trade missions)
Hittite
3 man chariot, iron swordsman, add Phrygian skirmisher
Ionian (Magna Graecia only)
Ionian cavalry, Ionian hoplite => add Ionian mercenary (version of armoured spearman) both with withdrawal and city taking bonus.
Minoan (Magna Graecia only)
Minoan archer => Minoan composite archer, Minoan bireme (as per Greek bireme)
Greek
hoplite (amphibious) => add phalanx (armoured spearman), both with city defense, melee bonus, add Greek trireme. How about Pericles for a leader ?
Thrace (Magna Graecia only)
Hillman (swordsman), Thracian Peltast skirmisher, add Thessalian cavalry (mounted javelineer)
Illyria (Magna Graecia only)
Kambsor spearman, Lembi (bireme) => add Liburnian (trireme, speedy, +withdrawal)
Scythia
Steppe rider => Steppe marauder (heavy horse archer), add mounted scout (scout)
Media (Near East only)
Wild Axeman, Mountain rider, add Hill archer (archer)
Goth
Goth 'Berserker', 'Huskarl' armored infantry, add mailed cavalry (heavy cavalry)
German tribes
Woodsman, Berserk swordsman, add German horseman (instead of chariot)
Gaul
Gallic swordsman, Gallic cavalier (heavy cavalry), add Gallic war cart (chariot)
Britons
Swift Chariot archer, Briton marauder, add Welsh longbowman (bowman)
Romans
Onager (catapult), legionary (infantry) => add Praetorian (armored infantry)
Hispanic Tribes
slinger => armored slinger, add Hispanic warrior (medium spearman, +forest defense, withdrawal)
Tartessus
Sounion javelineer, Iberian mounted javelineer, add Iberian swordsman (medium, +vs. heavy units)
Carthage
Libyan mercenary spearman, Carthaginian war elephant, add Numidian light cavalry
Kolchis
Kolchis war chariot, Kolchis marauder, add dromon (fire galley) - arbitrary choice but an argument could be made for it here, anyway the ship should be represented.
Dacia
peltast, Falxman, add Getae light cavalry
New civilizations:
add India, Asoka
scythe bearing war chariot (+ strength on plains), Indian war elephant, Indian bowman, hero: Chandragupta (elephant)
add Scandinavia, Canute
Seeax (axeman, melee, amphibious), longship (bireme, carry 1 unit, travel up rivers and 4 squares from shore), Varangian (infantry), hero: Beowulf (seeax)
add Gaelic tribes, Robert the Bruce
Coracle (transport galley, travel up rivers and 4 squares from shore), Highlander (swordsman, blitz, +forest, +hills), Schiltron (heavy spearman), hero: Wallace (highlander)
add Parthia
Parthian heavy horse archer, Parthian cataphract (heavy cavalry), Pontic spearman (+ hills defense)
hero: Parthian heavy horse archer
add Bactria
Bactrian mounted lancer (medium cavalry), Bactrian camel rider (heavy horse archer), Bactrian wagon train (provides mobile defense, reduces unit maintenance). hero: Bactrian mounted lancer
add Slavic tribes
Bulgar mailed cavalry (heavy cavalry), Slavic axeman (+city defense), Wagonberg (mobile defense)
hero: Slavic axeman
add Xiong-Nu
Yuezhi horse archer, desert rider (light cavalry, + desert defense, movement 3, blitz), Camp (settler, makes mobile town). hero: Modu Chanyu (desert rider)
add Ghana (Mali)
Skirmisher (archer), Sahel spearman, Touareg camel rider (light cavalry) all with desert defense
hero: Touareg camel rider
add Arabia, Saladin
Camel Archer (heavy horse archer), mounted swordsman (light cavalry), Hashashim assassin (spy) can assassinate great generals, spies, even great people. hero: Khalid ibn al-Walid (mounted swordsman)
Optional:
add Armenia (in Near East scenario only)
Pontic spearman (+ hills, cities), Armenian cataphract (medium cavalry), Armenian composite archer
hero: Pontic spearman
add Macedonia (in Magna Graecia only)
add Companions (heavy cavalry), Argyraspides (armoured spearmen), Hypaspists (swordsmen)
hero: Alexander (Companion)
add Israel, King Solomon (in Near East scenario only)
Israelite slinger (hills defense, +movement, 2-4 first strikes), Levite warrior (medium spearman, +vs. cities, hills defense), Maccabaean swordsman (hills, desert defense), hero: David (slinger)
More Generic siege units
Ballistas: strength 3, 1 ranged attack/turn.
Trebuchet: strength 5, reduce defenses 25%/turn, or 1 ranged attack/turn
Sappers: can build roads, forts, bridges or reduce defenses
colonist: settler with defense ability as per basic spearman, can build improvements as per worker, and after founding city leaves worker.
Wagon train: reduces maintenance costs for up to 5 items in square. Provides medic 1.
Siege quinquereme: as per quinquereme but no units, 1 ranged attack/turn at 4, or reduce defenses 20%/turn.
caravel: combination trade barge and transport (2 units), ocean going = dhow for Babylon, Egypt, Nubia, Arabia, Persia
Special:
modify Great general: provides one time 20 XP evenly distributed to units in hex. Also can be moved between units. Adds combat 1 and medic 1 to any unit attached to.
Great Admiral - similar but for ships
Barbarian warlord: Attracts any barbarian units encountered as adherents. Adds combat 1 and medic 1 to any unit attached to.
Hyenas, replaces wolves in the tropics
Treasure Hoard (earned): reduces maintenance costs by 50% for armies within 1 square. Can be used to bribe enemy spies or great generals.
National hero unit: Centurion (infantry), 3 allowed.
Military Wonders (mid to late game):
Helepolis (with advanced siege tech): 'city taker', 2 ranged attacks/turn at 5 as per trebuchet, or reduce defenses 30%/turn, carries 5 units as per siege tower, collateral damage, no withdrawal chance
Tesserakonter (with shipbuilding tech): 10/4, 1 ranged attack/turn at 5, carries 5 units, collateral damage, no withdrawal chance
Marian reforms (with bureaucracy civic and enfranchisement): 10% reduction in military support costs
Warrior Caste (with athletics, oligarchy, mysticism): units from that city gain 1 extra experience level
Famous Horse Stock (with selective breeding): cavalry gain +1 movement
Ishtar Gate (with corvee civic): City walls grant 10% extra defense value
Acropolis (with aristocracy civic): forts and citadels grant extra 10% defense value
vorpal blade (with steel): all iron based units reequip with steel
Other wonders:
Olympic Games (with organized religion and athletics): adds culture points, raises regard of other nations