Thanks for all of the replies. And yes, all will work if you want to Hide the Participants block:
1. If a user is enrolled in a course other than the one you want to be "anonymous", they can access Edit profile from there.
2. If not, you can give them instructions to click on their name to go to their profile.
3. Users can create anonymous accounts.
And, of course, you want to hide the Recent Activity and Online Users blocks also since they show "user info."