The only way to do this currently is two define two separate screens, each with their own background. The easiest thing to do is design one screen, then copy it to a new screen and change the background on the copy.
I'll assume you now have a screen called "Odd" and a screen called "Even".
In the Screen Sets dialog, you should have only the Default screen set. Add the "Odd" screen to it and set the screen display time to some high number, like 36000 (10 hours). Create another screen set called "Even rounds". Add only the "Even" screen to it, and set the screen display time to 36000. Press the Conditions button. In the "Stage" section, select "Round" and enter the word "even" in the input.
