Let me divide my comments into two areas of music: serious (by which I mean "classical" or "art" music) and popular (by which I mean commercial/everyday music for the masses). Serious music broke away from the "western orchestra" sound with such experimenters as John Cage and Philip Glass, both of whom saw any sound as music, even so-called silence. Arvo Part from Estonia moved away from the classical sound also, but used instruments and the human voice in new forms to express emotions beyond the romantic (Lizst, Faure, Ravel, Chopin, etc.).
In popular music, rap moved past the lyrical melody stage to an emphasis on rhythm/percussion and impromptu words (simulated), and thereby expressed social unrest, deviation from maudlin "love" impulses, inner city life, and post-modern impulses. Music, and in fact all art, seeks justification two times: at the creation (where the artist expresses) and at the reception (where the listener, viewer, receiver "feels" a reaction to the creative product).
New ways to express feelings through music are not much different compared to old ways to express feelings through music. Expression occurs more through the melody and sound of a song rather than lyrics. If you listen to classical music you can see just that. There are no words, but the song still tells a story and makes you feel something about it. A song speeds up and it may make you think of a quickened heartbeat, an intense occurrence, or feeling rushed. When a song slows down and gets quiet that conveys another emotion. Another good way to see that the sound of the song rather than the lyrics displays a lot of emotion is to listen to music in a foreign language you don't know. You can really hear all the subtle tones and gestures that are composed into the music. Some techniques to display specific emotions are in "the subtle changes in timing and volume that musicians make when playing."