There are also the Hashashin (more popularly known as the Society of Assassins), a secret society in the Muslim world around 1100 AD, run by a fellow named Hasan ibn Sabbah. Their followers were generally placed as moles in various governments in the region (and, later, around the old world). They would lay dormant for years, and then strike when orders were given, killing a high-ranking official without fear for their own lives. In game terms, it would probably be easier to make them an early spy unit, that can "assimilate" into a rival city, causing perhaps a production hit or some other negative effect when they're absorbed. (
http://www.unexplainedstuff.com/Secret-Societies/The-Assassins.html)
You also have the Knights Templar, who would actually make good counterpoints to the Hashashin. Perhaps they could be the same as your aforementioned Crusader units, except with the ability to spot invisible units?
The Israelis have the Mossad as a modern unit, but that's more of a secular unit than a religious one. Perhaps a more appropriate unit would be a Himayar Horseman unit (
http://www.geocities.com/jewishwarriors/yemen.html).
For Hindu, I would choose the Rajput (
http://en.wikipedia.org/wiki/Rajput) - mounted swordsmen who would never retreat from battle.
Buddhism is easy - I would probably choose Shaolin Monk as the UU... which would act as religious units (able to spread religion to other cities, but with lesser success than the missionary units) but they would also be able to attack and defend.
Confucianism is slightly tougher... the Samurai is the ultimate embodiment of Bushido (the "Way of the Warrior") but it's already present as a UU in the game. Perhaps rather than a UU we could have a Bushido Promotion that's only available to Confucian units... essentially allowing you to create Samurai Archers, Samurai Horsemen, Samurai Pikemen, etc.
That leaves Taoism... that's a toughie. The best thing I can think of is a Wudang Priest, which is a "fighting" style akin to Tai Chi. It's purely defensive in nature. I'd give them a high power, but the inability to initiate attack. They would be good for guarding workers or settlers or other lesser units.