Hey, I'm the author of Good Morning. I just released a new version that should fix the beeps. I don't own a Droid X (just the original Droid), and there's not much information on the net about what causes the beeps, so finding the problem was difficult, but here's my best guess:
When the text-to-speech service is set to use the Alarm volume and doesn't get properly killed (and it's still running during a phone call), certain phones (Droid X and Droid 2) will beep every 10 seconds.
Only 2 apps have caused this so far: Tasker and Good Morning.
Anyway, hope this helps!