I'd suggest that the Turks should have Religious as a characteristic. Religion was a cornerstone of their initial appearance in history - they came to Anatolia after they converted to Islam, and were kind of like the shock troops of Islam.
The Ottoman Emporer became the Caliph, the head of Islam (kind of like the Muslim pope), and they pushed their religion heavily. They brought Islam into the Balkans, which is why the Albanians today are Islamic.
Although there is some argument for making the Expansionist, in fact they really expanded more by war than exploration. They never got much of a navy going and completely missed out on the age of discovery, even though it happened during their golden age.
So, I would vote for Religious/Mililtary.
Janissaries are, IMO clearly their UU. Yes they used cannon to good effect, but this is essentially the same as them having discovered the advance and built those units before their neighbors did.
The Janissaries were unique to the Turks, and were a key to their success. The Ottomans (only one of many Turkish kingdoms at first) developed a caste of warriors who were recruited from Christian families, and raised from a very young age for service. They were the key to the Ottomans' conquest of the rest of the Turkish people in Anatolia, their expansion into the Balkans, and their defeat of the Byzantines. Their emergence also marked the beginning of the golden age.
I would make them a musketeer (or whatever the early gunpowder unit is, I don't have Civ3 yet), with no resource or support requirements, to reflect how they are recruited, but no attack/defense/move bonus.
and im 
