In -07 I went ´around the middle´ (Izmir-Konya-Malatya-Tatvan) to Lake Van, then turned north do Dogubayazit and Gurbulak-Bazargan border (that´s the biggest and northernmost one, I believe). But that was during the last days of October, and it was quite cold already. I hate to think what it would be like now... and on a streetbike!!
I´d probably try to go as far south as I could, maybe even follow the Mediterranean coast as far as I could (but that´s a very long detour – and whether or not the Black Sea coast would be an alternative, I dont know)...
and then I´d also try to find out, what routes there are in the east, that might have a little less elevation. I´m not sure, if there really are any such routes, because the whole eastern Turkey is quite mountaineous. Our route was very often about 1km above sea level, and the altitudes generally kept rising, as we went further east. But also remember, there may be other military-related troubles in the southeast.
I remember that someone also mentioned a train between Ankara and Teheran, that allegedly was a possibility to transport a bike... but sorry I don´t know more about that.
edit. Also check out some of the web´s weather sites (like BBC´s weather service, or accuweather.com) for forecasts for the areas you plan to go. And remember, those are usually for towns and cities, and when talking about a mountain area, the passes may be a lot higher and therefore colder. Our highest pass was maybe 2,5 kms above sea level.