From my list of games I choose one which is "My Move" and I wish to play.  After I submit my move another one automatically comes up.

If I have another "My Move" game, is there a way to jump directly to the next game, without making a move on the one presented to me?

Sometimes, I just want to wait on one game if I have the time.

That's the default behavior--after you move, the next ready move comes automatically.  Now, to stay on a particular game, you have to change your Online Games settings.  Go to your Online Chess, then click Settings.  The Preferences - Online Chess appears.  Scroll down to Move Preferences, and choose an option at the After Moving text box by clicking the drop-down arrow.  Choose Stay on this Game, and then click Save Preferences button at the bottom of the page.

Now, when you choose a game (from My Games), you'll stay with it even if you submit your move. 

 Exactly! You currently can only go to another game if...

1. You submit a move.

or

2. Go to your home, then select another game.

A "Next Game" button would be fantastic.

As for Charlie91, I like Ubuntu too! 

 But I don't want an action "After Moving"!  I want an action without moving.  I know about the drop down box!  But I see a game, I don't want to make a move,  I simply want to advance to the next ready game.  Therefore a next game button down beside, resign, draw, etc.  That would be handy.  

I'm not saying there isn't a way, as Reb explained... However I am suggesting a more convenient way.  Think not "work around", think "fix".