Before my poly adventure began, I was fairly religious and belonged to a church (that turned out to be a crazyyy place, even by evangelical standards) but there is one thing in particular that was "preached" that I took away. On the subject of conflict resolution.
Sometimes we hurt people, sometimes we mean to, sometimes we don't. They may be crazy and overly sensitive. Regardless, if some one comes to us and is hurt by our words or actions, one of the best ways to restore friendship is to simply say, I am sorry I hurt you. or something to that effect. To apologize that your actions caused them some pain. I've found that that alone can do so much that the issue often dissipates. Especially if it was something minor to begin with. Perhaps then if they've hurt you too, you can ask them to apologize to you. I know that was/is a hard mental transition for me because I love to be right. And if I think I'm right then I sure as hell don't want to apologize for something that's probably your fault anyway. But that kind of thinking has never helped me mend any relationship. So I've been working on it.