There is an easy way of achieving that, though it requires headphones.
I've been using a
Turtle Beach DSS2 throughout most of the 360 gen, and I'm now using it with the XB1 (I do almost all my gaming with Sennheiser 'phones - I simply prefer it, no matter how decent my vintage Cambridge speakers still are). With that you can just insert a 3.5 jack and connect an MP3 player (very occasionally I connect a laptop instead).
I sometimes listen to podcasts whilst playing Elite, and so I disable the supercruise music, but usually leave everything else on. The DSS2 has a volume which can also be clicked to mute game audio, but the auxiliary feed is unchanged - volume control remains on the device itself. Having independent control like that is great, as you don't need to dive in to any menus to pause music/podcasts, or adjust volumes.
Oh, and yes, the DSS2
is/was ostensibly a surround-sound approximating device, but I've not used that gimmicky feature for years - I stick with the much cleaner, sharper stereo. The device also has some EQ presets, too, which can be cycled through at the touch of a button on the side. All in all it's a great little device, and whenever this one dies I'll be getting a replacement immediately.
/edit
Ah, didn't see that post about the XB1 update. Regardless, for me the DSS2 solution's
vastly superior, as all control is completely independent/external of the game and console (the game or console could crash, but the music/podcast won't be affected) - I can end a gaming session and just continue to listen to whatever I was, with no interruptions.