As I see it, there are several issues involved.
The problem with a lot of contemporary Christian music is that it is shallow at best, and often theologically inaccurate. It is no wonder some folks get a little distressed by it. It does seem that the older hymns (while some may have been written to the tune of contemporary tunes of that day) were written by older people, who had a better grip on theology, and who had some experience in life, and particular with suffering (e.g. Fanny Crosby).
I think there are a number of people who dislike certain types of music or worship, not because it is wrong, but because they are too proud to do it. Raising one’s hands, for example, may be something which I refuse to do because I am too proud to do so. Of course there are other reasons why I may choose not to do so, but pride is one possible explanation. I never cease to be amazed at the different ways people define “worshipful” music. For some, this means singing slowly, for others fast; for some it means quietly, for others loud and enthusiastic.
Our brothers and sisters in the African American churches certainly sing different songs than we do, and so do those of other cultures. Around the world there are many different styles of worship, and thus of musical expression for worship. Much of what we may hold to be “sacred” may simply be that which is familiar to us, and that which is a part of our culture. But I wonder if the reason why God is saving people from every language and culture is that He wishes to be praised in a wide variety of ways, with a wide variety of music.
In a church I think this matter needs to be dealt with first in principle (e.g. “putting up with one another”), and then in practice. I think any changes need to be explained in advance, and brought about slowly. The one thing I would strongly recommend against is having two different worship services, one a “contemporary worship service,” and the other a “traditional worship service.” This sets the stage for a church split, and it lets saints off the hook too easily. This lets each group do its own thing, without “putting up with one another,” without making concessions and sacrifices. I would say that all should worship together, and that there should be a balance of music, with each group giving ground to the other, and not just demanding their own kind of music. Heaven will be one large worship time, so why should we segregate our worship on earth? I believe that the unity we express in our diversity is very significant (see Romans 15:7-13).
The leadership (pastor, elders) should avoid taking sides if at all possible, unless one side is completely right, and the other completely wrong. If those stating their position are required to cite chapter and verse for their views, there will be a lot less “opinion.” If the “conviction” card is played, we should be reminded of what Paul says about convictions: “Keep them to yourself, don’t argue about them, and don’t divide over them” (Romans 14).