Is it proper for a church to sing worldly songs to praise God?
In the words of Paul, all things are lawful, but not all things are profitable. By this we mean that the Bible doesn't prescribe a specific style of music (or any music at all) as part of our worship of God. Instead, the Bible teaches we are to worship in Spirit and Truth (John 4:23-24). Specifically, we are called to worship by allowing the Spirit of God to direct our attention and needs toward God in all that we say and do, and we are to conduct our worship focused on the teaching of and preaching of God's word.
Therefore, if we choose to use "worldly" music in our worship services, we must be careful to select music that adheres to these requirements. The words we sing must reflect the leading of the Spirit and reiterate truths found in God's word without being trivial or offensive. Obviously, it will be difficult to find worldly music that meets these requirements, but when we do, such music could be a useful part of worship.
The content of the words is the key issue in deciding whether music is acceptable for worship, but virtually any musical style could be acceptable so long as it supports these same two expectations. Today we can find many worship songs using Biblically-based lyrics set to contemporary musical arrangements. We can still find traditional hymns in many churches as well. The style of music is not a principle concern in the course of worship, but the words we use and the effect it has on our ability to worship are very important.