There's a lot of incorrect information in the responses. I registered to post some easy instructions in case others have the same question and incorrectly assume that all the "no" answers mean you can't do it. You can. It's fairly easy. Alexander Kononenko had the right answer but didn't really tell people the necessary details.
In my case, I'm using the apple music app from an android phone. If you're using the apple music app from an apple phone, you can just cast to chromecast directly from the apple music app. The cast button is conspicuously absent from the Android version of Apple Music.
You can cast audio(or video) to a chromecast device from an android phone even if the android app doesn't directly support casting. You do it with an app called "Home" from Google.
Step 1. Run Home (aka Google Home). It's on the Play Store if you don't already have it installed, and it's free.
Step 2. Touch the "Account" button. On the main google Home screen, it's the bottom right button. The button looks kind of like a circle with a drawing of a person in it.
Step 3. Touch "Mirror Device" from the menu that came up after step 2
Step 4. Click "Cast Screen / Audio". It's the only button on the screen, so it's easy to find.
Step 4. Select your chromecast device. If the chromecast isn't listed, you might not be signed into your home's wifi. If you've got the device already setup in google home, it might be listed more than once. In my case, I'm casting to an Onkyo TX-RZ820 stereo with "Chromecast Built-In".
Step 5. Run the Apple Music app and play the song of your choice!
Actually, you can replace step 5 with anything you like. After steps 1-4, your phone will be using the chromecast for audio/video output from your phone. Any thing you play on your phone will also play on your chromecast. If you're playing video, you might need to tweak the settings on your phone so that the audio and video are in sync. You can google for how to do that separately.
Anyway.. as you can see, there's no hacks, third party apps, drm removal or any of that nonsense required. You can play anything on your chromecast that you could play from your Android phone. That includes music/video played through Apple Music.
Personally, I prefer Google Music over Apple Music but Verizon is giving away Apple Music subscriptions for free (if you've got a Verizon unlimited data plan) so I'm moving to Apple from Google for music. Free is for me after all.
Hope this helps clear up some of the nonsense floating around (and in this message thread). Enjoy your music, and don't worry about DRM. In this case, all the DRM in the world is a non-issue. This method isn't illegal. This method is fully supported by Google (since it uses a Google Chromecast and Google Home). Apple doesn't specify what kind of speakers or listening hardware you use in their EULA, so you aren't violating the terms or anything of the sort.