Are many businesses likely to accept US dollars or Euros in Turkey?

Euros no, they are not yet part of the European Union. Possibly/probably to the U.S., so long as it is not difficult for them to give you change and it is a more touristic place they can take them. You can always try with U.S., but have some Lira as back-up.

In touristic towns, yes, they'd accept US in all shops. But if you think you can get accustomed to the many zeros of the Turkish liras(lol), you may choose to change your dollars in exchange offices as the rate in shops may be slightly lower. Exchange rates in hotels can be lower too than the exchange offices outside. In general you won't have any problems in Turkey if you have US dollars in your pocket.

However you are going to end up paying more if you pay in USD. Its better to pay in local currency.

I had to laugh at Nurale's comment about the zeros in the Turkish lira! When we arrived in our first town in Turkey, we were approached about a hotel. The rate was quoted in Turkish lira and the amount was in the millions. I wish we had a picture of our expressions as we tried to convert it to Canadian dollars. Then there was the small silk carpet we bought - I'm surprised the salesman's calculator could accommodate that many zeros!