Don't forget you can also set which page you go to after you move, so you can choose to return to 'My Games' each time instead of going to the next waiting game. That way, if 'My Games' is sorted by 'next alert', then you will see which game you should go to next.
Another option is to use a tabbed browser, such as Safari on the Mac or Firefox anywhere, and open all the urgent games in tabs before you start making moves.
I must admit that I'm not in the same league as Harri for number of games, so maybe what I do won't work for him. By the way, I'd say Harri is an ideal opponent - plays fast whether he's winning or losing and doesn't have massive ratings swings. (Of course the fact he gave me two good games without actually winning was nice too