Obviously, a function continuously running within the actionPerformed method will effectively disable the rest of my GUI.
This is a valid observation and shows that you have understand the fundamental rule when working with Swing.
Your game is most likely event driven (correct me if I'm wrong) so the action performed by the button should just set the program in a new state, waiting for further events. This is nothing that should be time consuming, and is typically done directly by the EDT.
Of course, if you want to do a fancy start-new-game animation, that needs to be performed in a separate thread, in which case you simply start the animation thread (I would recommend using a SwingWorker though) from within the actionPerformed method, and then return.
In code, I imagine it would look something like this:
go.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
// Remove the menu components
someJPanel.removeAll();
// Show the game table
someJPanel.add(new GamePanel());
someJPanel.revalidate();
someJPanel.repaint();
// done. Wait for further user actions.
}
});