"Er hört während seinem Dienst Musik."
"während" is a genitive preposition, so the sentence would correctly be: "Er hört während seines Dienstes Musik."
Am I right?
Yes, you're right. (Although you'll sometimes hear the dative construction used by native speakers - especially on the internet and by teenagers -, it is considered wrong)
The Duden doesn't even consider the dative construction as colloquial. It should be avoided even in informal speech. [I know you know, I stress thist just to counter confusion among non-native readers of this thread ;-)] But I agree that not all native speakers are that picky about it.
Just chiming in to say that "während des Dienstes" and "während dem Dienst" (if you're not too fussy) sound slightly more natural.
Thanks. For my ears, 'während dem Dienst' sounds almost acceptable in colloquial speech while "während seinem Dienst" is just awful ;-)