I personally would never use blinking. I'd instead use colouring - make it red or something like that when there's output, then make it black again when the tab is selected. Since
Java 6 JTabbedPane has method setTabComponentAt which allows you to set a simple JLabel for that one tab, letting the default UI work for the others. There's no need for a timer then; keep a reference to this JLabel and simply change its foreground colour when needed.
Edit: or use setForegroundAt. I never knew this method existed. Thanks Darryl!
And Tarun, you clearly have not read the concurrency in Swing page Darryl linked to, or you wouldn't be using a regular thread.