They have live bands at Karmylassos every Wednesday night. That's the new place on the bends by Muzzy's and the Istanbul. It's Turkish music, but modern rock and pop stuff, not traditional.
The only time that the Istanbul has live music, to my knowledge, is when there's a wedding being held there.
Karvansary is your best bet for live Turkish music. Plus, if you book through an agent, you can get free transport arranged.