Update 7/1/15: Beats trials are no longer available, so if you didn't have one before now, this doesn't help. Also another option for Spotify has emerged, called Exportify, which works in a similar fashion for migrating Spotify playlists, either one or many at a time.
Spotify and Rdio users can make their Apple Music trial a smoother experience by having those carefully crafted playlists available to them from the beginning. According to the Beats Music FAQ, when you sign into Beats Music for the first time after Apple Music launches, you'll get to transition your Beats playlists over to Apple Music. Since there's no certainty of any import tool working with Apple Music, or this third-party utility continuing to function after Apple Music launches, right now it is a good way to get set up with your custom playlists with a minimum of hassle.
There are some caveats here: You can get a free 14-day trial of Beats if you've never signed up with your email address for a trial before. Also, the site is under some serious strain right now, since lots of people are trying to do this today. What's important is that you get your Beats trial set up ASAP* (This is no longer possible, trials aren't available on Beat anymore), then you can hopefully hit the Beats Importer later, after the stampede has died down a little. Since you don't need to actually launch Beats Music to hook it to Beats Importer, you should theoretically be able to do this as long as you don't launch the Beats Music app or log in on the website, both of which will prompt you to move over to Apple Music.
Once you connect Beats Music to Beats Importer, then you can choose which service you want to import from. Your options are: CSV file, Spotify, XSPF Playlist file, iTunes Export, Rdio, Windows Media Player ZPL, and even Beats itself.
Front page of Beats Importer, waiting to connect.
I was walking through this process to get screenshots of everything so I could show you the process, but the site is extra popular today so it is having a hard time loading, as I mentioned above.
Since my example is Spotify playlists, here's what happens: After hooking up Beats Music, you choose the import type. I'm going with Spotify playlists, since that's what I have. So I click the button to connect with Spotify.
Beats Importer asking to connect to Spotify.
After giving Spotify permission to hook up to Beats Importer, go back to the main Beats Importer page and you'll see the menu under Spotify which now shows your playlists. Choose the one you want to import (you can only do one at a time), and hit submit:
Beats Importer showing Spotify playlists to import.
Once you give it your playlist, and everything gets imported, presumably you can now access that playlist in Beats Music. This is the part I can't show you since the site keeps crashing. When you have successfully imported one playlist, you can move on to the next and figure out which ones are the ones you want to use with Apple Music.