I just started therapy at a program at Beth Israel Hospital. Sessions are only $15-$40 depending on income. The catch is that the sessions are recorded so the third-year residents doing the therapy can be evaluated. You can do the program for a year or more.
They have to pre-screen you first and I couldn't really tell what they were looking for. (Am I crazy enough???

) But, if somehow you don't qualify, they give you information on where to go or send you to another program that they think you would benefit from.
I am a woman with a woman therapist and didn't request a therapist of a certain sex, but I am sure that's something you could ask about. The whole process has been very easy-going and open.
I am not sure if it is a seasonal thing or if they are still looking, but it is worth a shot.
I can't find the info for the person I originally spoke to, but I did find this link which seems different than what I am doing, but is a good starting point:
http://www.wehealnewyork.org/studies/briefpsycho.html.
Next time I go to a session I will ask my therapist about whether or not they are still looking for people and who to contact.