Seat61 is a great site all about train travel.
Check out Train Travel in Turkey.
Though I've never been to Turkey I can't see why you wouldn't be able to travel by train in the winter months. So many rely on train travel throughout the area and generally countries with adverse weather conditions tend to have found ways to deal with adversities. I think you should fair well during the winter months as long as you're gear is adequate.
Couple of things to note;
It's low season and some hotels and such shut down. The other thing I noticed is alot of people warn you need to be aware whether your room price includes 24 heat or not. So be sure to check. I've also read that flights can be very cheap during that time so maybe an option...personally I think train travel is the way to go!