As great as Google's own Play Music app is, your MP3 library is restricted to Google's own app. There are much better music players that give Google Play Music a run for its money, but duplicating music files on your device for this reason alone is a waste of storage space.
Note that while I'll be showing you this mod on my rooted Samsung Galaxy Note 3, is will work for any rooted device running Android 4.0+.
When you first open the app, you'll need to provide it with root permissions, so hit Grant when prompted.
Start by tapping Account to chose your Google account, then tap OK. A popup will appear asking if you want GMusicFS to have access to your account—tap OK again. GMusicFS will perform a scan of your Google Music library and begin syncing. Depending on how many MP3s you have stored in the cloud, this could take a few minutes.
Now that your collection is synced, you'll need to configure the streaming quality for playback, so tap the tab for Settings in GMusicFS to get started.
Change the value in Stream buffering -> Buffer length to 30s and Pre-buffering length to 1s. Also check the box in General -> Start on boot so your library is always available when your turn on your device.
Now just launch your favorite preferred music player and you'll see your MP3s available—just as if they were locally stored on your phone.
That's it! Now your Google Music library is free to be played in any app. Give it a try and tell us if it worked for you in the comments section below.