As mentioned already, Blazemongers solution is failing to remove quotes (e.g. returned by Firefox). Since I find Rob Ws solution to be rather complicated, adding my 2 cents here:
My two cents.
None of these solutions work if the background-image property is overwritten in CSS by the background property.
Let's say you have the background-image property set and you don't want to see it in the browser, you just need to get the value in jQuery for some purpose.
So, if you have something like this:
Then $('div').css('background-image'); returns none instead the URL.
If you need to keep background white and also to get the background-image URL value in jQuery then replace the background with pseudo-element, like this: