The Custom Design upgrade has a colors feature that let's you change the base background color, but not necessarily the color behind the content container. Which colors change depends on each theme's setup. You can test out the colors feature by going to Appearance → Custom Design → Colors.
If you pick out a color palette and the content container color doesn't change, you can still change the background color behind posts by using a little custom CSS. Here is an example for the Ever After theme:
Update the color code to a color of your choice.