A wise pastor once said, "Hymns are small sermons."
We should ask ourselves: if we wouldn't allow a particular teaching to be set forth in a sermon, why would permit it in a hymn?
The issue isn't usually (though there are exceptions) with hymns that have gone through a doctrinal review process, and are found in hymnals. It's with the ones folks want to use that they've heard, and think would be cool to use in church.
If a song didn't make it into the hymnal, there's often a good reason why it didn't. Of course, new hymns are being written all the time. Some of those will -- at some point in the future -- be included in a hymnal. But before using an untested song to be used in the Divine Service, ask: would I say this in a sermon? Or would I find this acceptable teaching if I heard a pastor say this in a sermon? And if not, why would we sing it in a song?
Just because a song has words like "Jesus" or "pray" or whatever doesn't mean it's good. Listen carefully. And when in doubt, leave it out.