I was reading an interesting discussion on DemocraticUnderground the other day. The basic idea being proposed was that we can debate and question facts and ideas, we can praise or criticize actions, but when we judge another's feelings as wrong, the discussion generally turns a turn into a useless emotional skirmish.
In order to understand another person, we have to learn that sometimes we have to just understand and accept their feelings as a starting point. That's what drives their beliefs and actions. Without understand their feelings, attempts to "rationally" argue them out of a position is usually an exercise in frustration. In most cases this means actually respecting their feelings. You may see things differently, have different priorities, and have different emotions about any given situation, but you need to understand that the other person has valid feelings about the same situation that are different from yours, often just because they give different priorities to conflicting aspects of the situation.
Admittedly there are some cases where respect for another's feeling may be inappropriate - cases where emotions are driven by anger, prejudice, or an overwhelming desire for revenge. In such cases having a civil conversation may be very difficult. The best you can do is to establish a personal relationship with the other person with the hope of something changing in the future. That requires you to at least understand and accept what the other person is feeling.