Updating GUI Components From A Thread

I've been reading up on how to update a GUI component from a Thread
and, as far as I can tell, I need to have my Thread update the GUI
component (JLabel in this case) by using a
SwingUtilities.invokeLater(Runnable) call inside the worker thread at
various points. An example is shown below (I realize there are
problems with this code - I'm just trying to demonstrate an
understanding):

    public void run() {
        while (true) {
            try {
                    // do some stuff and set the boolean variable
"failed" based on the work done
                    SwingUtilities.invokeLater(statusUpdate);
                }
            } catch (InterruptedException e) {
                e.printStackTrace();
            }
        }

        return;
    }

In the constructor of the worker class, I defined statusUpdate like
this (status is a class that extends JLabel):

statusUpdate = new Runnable() {
            public void run() {
                status.updateErrorStatus(failed);
            }
        };

What is confusing to me is that the class that is doing the
"work" (the thread class) needs a reference to a GUI component (or
possibly more than one GUI component if there are multiple views that
need updating based on this thread). Maybe I'm getting myself
confused, but that seems like poor design. Am I thinking about this
the wrong way? Is there something I'm not understanding?

Thanks for any explanation.