1. There have been terrorist attacks throughout Turkey in the last many years. The most recent incident in Istanbul was outside the American Embassy in July or August 2008 when four people including some policemen were killed. There were also bombing incidents in Izmir, Istanbul and other places. So try to avoid places where there is a large congregation of people, of course this is not easy (if there are few people, you will probably not go there anyway).
2. There are some (although rare) aggressive touts in the retail and restaurant businesses who are too persistent and do not accept no for an answer. This has happened even in the small tourist town of Kusadasi off the Aegean coast where a small shop selling turkish tea and other things asked for an exortibant price for the tea and when offered a lower price, turned abusive. Such behaviours should be reported to the tourism police for prosecution.
3. Beware of dehydration, sunburns and sun strokes especially when you are outdoors visiting ancient ruins in sunny and hot weather. There are no shades at these sites and you usually have to walk for hours. Always have bottled water with you and drink lots of it. Also, use sunblocks and umbrellas, hats or long-sleeves dresses.
4. Avoid uncooked food or opened fruits.
5. The following is more an advice than a precaution. Places such at Aya Sofia, Basilica Cistern, Topkapi Palace, Kariye Museum and the Blue Mosque do not allow use of tripods inside the buildings. So, do not bother to bring them along when visiting these places.