After some Googling, I finally figured out how to do this because I'll be damned if I'm paying Microsoft an additional $10 a month to stream my own music to my xbox while I play Forza.
It turns out the Xbox Music app can act as a dumb DLNA receiver. Using the free Android app called Skifta I was able to turn my HTC One into a DLNA server, and from there I could send any local mp3s on my phone or things in my shared folder on my networked PC to the Xbox Music app, however, I've run into 2 issues that are keeping this from being an ideal solution during gameplay:
1. Every time a new track starts playing, the Xbox pauses the game and switches attention to the snapped side of the screen. Meaning every 3-5 minutes I have to say "Xbox: Switch" (often multiple times until the amazing new Kinect recognizes it) and then unpause the game. Pretty clunky, pretty annoying.
2. There does not seem to be a way to play music during gameplay without it being "snapped" to the side of the screen. I don't need the track info taking up a quarter of my play area! I don't get how this is a better user experience than having all of this stuff accessible in the mini-guide.
There are a lot of baffling UI decisions with the Xbox One that make it feel a step backward in functionality that I can only hope get fixed in a patch at some point.
Luckily my HTC One has loud, ok sounding speakers and I've just been playing music through that as a compromise until I can figure out a better solution. If anyone knows a better way to do this though, I'd love to know.