Originally posted by Russ
Then, if you use IE, hold shift then click each game in turn.
Each will pop up in a new browser - solving your problem.
With Mozilla it's even better: right click the game you want to play and choose "Open Link in New Tab". No new window, everything in the same window.
After the second tab is opened, you don't need the right click, all you have to do is to drag the link from the "My Games" page to the tab of the game you just played a move, and it will be opened in that tab.
You can even have more tabs opened, one with "My Games", other with the game you are playing a move, and other to your favorite Forum, and other to the "Open Invites" or "Find Player", or in "Help", or in http://chess.about.com/library/ble50ndx.htm?PM=ss13_chess.
After all your favorites tabs are openned, you can bookmark the set of tabs, instead of only a site. So, when you click on that link in your bookmark, all that set of tabs and associated pages will be opened to you.