Note to self: Feelings matter.
It's not necessary that you think their feelings are rational. It is necessary that you find out why they feel the way they do (let the other person use their words and just listen), and then address the feelings.
Acknowledge that the feelings are real, and that the person has the right to feel the way they do, and then ask the person what precisely would make them feel better. You might not be able to solve the problem exactly the way they would prefer, but you can at least move towards an amicable solution.
Now, I just have to remember this when it counts.