I wouldn't wait for a few sessions, I would find out before laying down a single dollar. You're within your rights to interview potential therapists before hiring them. If they refuse to interview, walk away. And during the interview, ask them specifically if they are polyamory-friendly. Use the word "polyamory." Whether they know the word will be the first test. Their reaction will be the second test. And a direct question whether they support it will be the final question.
Agreed, but in my case, I liked the therapist initially and felt comfortable with her after having shopped around for a therapist for a while. Then, a few sessions into it, when we got deeper into my issues, I felt like she was saying unhelpful things and/or wasn't understanding my situation. But at that point, I felt too invested or too guilty to change therapists.
I was also concerned that it was my problem if my therapist wasn't telling me what I wanted to hear. I thought that I had to face uncomfortable truths or something.
I was also severely depressed/unstable at the time and could not have handled interviewing a whole lot of therapists.
Oh, and the biggest issue was, I didn't know I was poly and would not have been able to use or articulate the word "polyamorous" in my search for a therapist. I was indeed dealing with the fallout of a non-exclusive relationship, and I'd had non-exclusive relationships in the past, but part of my problem was that I had no idea why these relationships hadn't worked out or what was "wrong" with me.
This particular therapist kept agreeing that she, too, didn't know what was "wrong" with me, except that it seemed like I wasn't looking for a "real" relationship. (Which is exactly what my ex had told me that plunged me into a crisis in the first place--that he had never considered our [4-years' long] relationship "real" because I had wanted to be non-exclusive).
I finally stopped making appointments with her, but didn't seek out a new therapist until months later. By then, I had learned all about polyamory and had made a lot of strides on my own in making peace with myself and the break-up. For my next therapist, I was much better able to articulate what I was looking for in therapy, and I found a really open-minded therapist who told me right away that she has experience treating people in open relationships.
I didn't even realize how out of sync with me that previous therapist had been until, in retrospect, I realized how much more useful my current one is.
So that's all I mean--even if you like a therapist initially, you may figure out later that you don't click well, and it's not too late to find a new therapist then.