I'd say US dollars. Certainly in Thrace and on the Mediterranean coast of Turkey, euros would be handier to have - many hotels and other tourist places will price things in euros. But you say you're not spending too much time in Turkey, and I'd guess that even if things go wrong with your ATM card, it would be much easier to rectify in Turkey (plenty of foreign banks like HSBC have a presence there) than in Syria.
By the time you get to Syria and Lebanon, you'll frequently see things priced in dollars. If you're staying in mid- to high-end hotels, it's not uncommon for them to actually request payment in US dollars. If you catch a private taxi from Damascus to Beirut, they'll usually ask for USD... I do find a stash of dollars a very handy thing to have when I'm in the Middle East.