I had pretty much the same question. After some research, I think that if you want to play the default system "notification sound", you pretty much have to display a notification and tell it to use the default sound. And there's something to be said for the argument in some of the other answers that if you're playing a notification sound, you should be presenting some notification message as well.
However, a little tweaking of the notification API and you can get close to what you want. You can display a blank notification and then remove it automatically after a few seconds. I think this will work for me; maybe it will work for you.
I've created a set of convenience methods in com.globalmentor.android.app.Notifications.java
which allow you create a notification sound like this:
Notifications.notify(this);
The LED will also flash and, if you have vibrate permission, a vibration will occur. Yes, a notification icon will appear in the notification bar but will disappear after a few seconds.
At this point you may realize that, since the notification will go away anyway, you might as well have a scrolling ticker message in the notification bar; you can do that like this:
Notifications.notify(this, 5000, "This text will go away after five seconds.");
There are many other convenience methods in this class. You can download the whole library from its Subversion repository and build it with Maven. It depends on the globalmentor-core library, which can also be built and installed with Maven.