This appears to be a common practice among musicians nowadays, and some rabbis apparently have no problem with it. However, there are those who say a non-Jewish melody needs to be somehow “rectified” before it can be made use of without danger (Prolific Chabad kabbalist Rabbi Yitzhok Ginsburg holds this way, I believe — you can check his website to find more).
One interesting hashkafic source on this issue is Likutei Eitzos, Music, which you can read for free in English online. Here are a couple quotes from it:
“The miserable wailing of the songs of the wicked does a great deal of damage because people are easily influenced by them. But when the Tzaddikim sing these songs on Shabbat they elevate and purify them.”
“A holy melody gives strength to the forces of holiness. But the music of the Sitra Achra, the Other Side, damages these forces and lengthens the exile. It makes people stumble and traps them like birds in a snare. Be very careful never to listen to this kind of music at all. The musicians and singers who produce it have no religious intentions whatsoever. On the contrary, they only want to make money or become famous. Listening to this kind of music can seriously weaken your devotion to God. But the melodies played by a truly religious, God-fearing musician can be very inspiring. They can strengthen your devotion immensely.”
“Man is endowed with an image-making faculty, which is one of the most powerful forces in his life, because it is through this that he forms his concepts. But because it is linked to the imagination, it is seriously prone to error. Good and evil are mixed up. But music played by a truly God-fearing player for the sake of Heaven has a wonderful power to subdue the wildness of the imagination, and the good is sifted from the bad. Such music has the power to lift you from depression and inspire you with joy. This is the way to develop a good memory, which means to remember at all times the goal of the World to Come and to understand the things God sends every day to draw you closer to Him. When you have a good memory, you are free of the deceptions of surface appearances.”