This doesn’t bother me, even when the tune in question is not to my taste.
What does bother me is when the singer exploits the text, and sings a song simply because one of the words sounds like his name, or similar childish reasons.
According to the Gemara, in such cases, ????? ????? ?? ??????? ???? ???”?, and this is the source of much suffering. Even ??? ???? was punished for saying ?????? ??? ?? ?????.
In my opinion, much of the “Goyish-sounding music” about which people love to complain is actually preferable to a lot of the songs these people prefer, if only because their lyrics are not inherently disrespectful of the Torah. It’s better to sing words of your own composition than to abuse the ??? ???.