I have an account with Ziraat Bank(next to Migros in central Fethiye) - they do normally make a small monthly charge, but if you set up two automatic payments for paying utility bills they are prepared to offer free banking - though you may need to negotiate to get this.
There is one male cashier there who's English is very good.
Just one word of warning - be prepared to queue! They close 12.30 - 1.30 for lunch...the quietest time I have found is about 12 midday.