I've heard the "House of the Rising Sun" melody used for other hymns in church. I'm not sure what the thinking behind that is, since most of these hymns already have their own melodies.
My guess is that those church's music leaders are huge fans of secular music, so they are simply trying to merge that with worship. Which probably skirts dangerously close to blasphemy.
Worship isn't for our benefit or entertainment, it's for GOD and GOD alone. Anytime worship gets contaminated by our own personal interests, then it's no longer worship, it's idolatry.
Unfortunately, many modern Christians (at least modern American Christians) are incapable of distinguishing between music and worship. I think the current "music=worship/worship=music" trend in modern Churches is one the great heresies of our time.
Worship is not something you conjure up from an emotional reaction to music. The inability to distinguish between emotional feelings and true worship sets us up for major Satanic deceptions, even within the walls of a church building. Yes, true worship can inspire emotions, but emotions don't necessarily inspire true worship.