I so envy my artist friends who can listen to podcasts, or books on tape, or sports, or even movies as they work. So efficient! Unlike many of my writerly friends, I can almost never write while listening to music with lyrics—I have no idea how they even do that—so obviously something with a competing narrative is right out.
For me, it's got to be almost entirely instrumental: mainly classical (chamber music usually works better than orchestral) or jazz or ambient. Keith Jarrett, Brian Eno, Debussy. World music works, too, especially if I'm writing something set in another country. Recently I've been listening to a lot of post-rock, as it tends to be light on vocals.
The past few days I've been listening to a lot of Bill Frisell, especially his solo concerts, or his duets with the great pedal steel guitarist Greg Leisz. Here's a similar show where they're joined by a rhythm section, and featuring the work of some composer named John Lennon who, judging by the evidence here, had a way with melody. (If he had any facility whatsoever with lyrics, he might've been a contender.)