Rav Belsky told me once that a non-jewish melody can be used for a jewish song as long as it wasn’t used for a love song. He also said that the kashrus of a song depends on the kind of middos that it elicits and excites. A song which evokes a feeling of hefkerus, or makes you want to dance wildly, is not kosher, no matters who composed it.
One exception to this seems to be the famous story about the old kalover rebbe who overheard a farmer singing a song of longing for his wife – the rebbe “purchased” the song and was mekadesh it, writing Hungarian words to the tune about longing for moshiach. I know the yiddish version, but while i can’t speak Hungarian, I’m told it’s amazing.