I spent some time this morning playing around with CSS to see if I could get something to work.
It seems like a path to explore would be to add some custom code to the Code Inside Header section of your site along the lines of this:
I hasten to add that this CSS is not correct - you will need to experiment with replacing the * with a more precise selector. The approach I was taking was to set the background color of one block, then use the Chrome debugger tools to explore what that did to the CSS on the page, then try to reproduce that selector in the site settings code. I didn’t get far enough to offer a solution but if you are able to experiment further with this approach, you might find it!