"What music do you listen to?"
I don't think so. I think it would sound about as strange as the equivalent would sound in English ("music what" instead of "what music"). The word order in Chinese is pretty strict.
Not that it's proper English, but it helps me remember Chinese more easily when I think of it this way.
The way I understand it, it mostly works if you put the interrogative (shen me) where the answer would appear in the similar declarative statement answer. So
Q: "You listen to what music"
A: "I listen to American music" (or whatever)
The English grammar difference between this and your suggestion of putting it at the end is that the English word "what" is sometimes a determiner (goes before a noun) and sometimes it is a pronoun. In this example it is a determiner and goes before the noun "music" and it does the same in Mandarin. But in the sentence "what are you called?" it is a pronoun standing on its own, and the Mandarin does something similar but it goes at the end of the question not the beginning (ni jiao shen me).