Shape Overlapping

I'm trying to draw a kind of background image, with bubbles overlapping, and I want the final product to be a vector with a transparent background.  I can draw it with a solid white background, like I started below, and I know that I can take each individual bubble, and cut the excess off, but I was wondering if anyone knows of an easier way to do it, where I can just remove the white for the whole piece at once, but not see the overlapping lines underneath, like I would see if I just make the background of the bubbles transparent?