Most of it is passively paying attention. It should not be a struggle, it's one of those the more you struggle and overintellectualize the less time you are focusing on paying attention and letting your hearing ability do its work it was evolved to do.
The other thing is this whole emphasis on accents is misdirected. Teachers do not place this excessive emphasis on accents, it is people who want to sound "authentic" which is not a very wise goal of language learning in the first place.
I do think that learning music can help a little, especially a sonically complex instrument like violin and the like.
(caveat: I'm way oversimplifying on my Saturday afternoon, but that's my tentative views on this that I would try to argue for.)