We must recognize that our "culture" whatever the factors that make up that culture, influences our leadership and our leadership also affects culture. In problem solving, it is always important to make sure that everyone's voice is heard but we must also realize that usually "making everyone happy" is not realistic and should not be the ultimate goal-but instead to come to a conclusion/resolution that best fits to address the issue.
In the LEAD Tier III Cross Cultural workshop we used Hofstede's Cultural Dimensions model and applied it to organizations or groups that we are in and then we walked through a collaborative multicultural problem-solving process. The process is as follows: 1) Identify the Problem, 2) Consider all different perspectives of those involved, 3) Identify challenges and opportunities within the conflict, 4) Brainstorm various approaches and strategies, 5) Turn your strategies into a solid plan of action, and 6) Identify expected outcomes.
This is a great model that I feel can be used to help solve any conflict big or small, multicultural or otherwise. As a leader, it is important to think about how to go about effective problem-solving and how to get those involved to see past their own points-of-views, which are naturally self-centered and get them to see the perspectives of others in the group. Once you have open-minds in a group joined with some empathy/understanding, I think you will be well on the road to better compromise. Now, this is much easier said than done and sometimes resolving conflicts may involve people being dismissed or people walking away, as that is life. However, a leader must be confident in their final decision and that that was the best possible one made for the betterment of the group, organization, etc.