The law that allows musicians to get paid is copyright law. As far as I know there isn't any other specific law involved (there are laws that work on top of, or hand in hand with copyright, like the Digital Performance Act, but BMI and ASCAP are basically copyright enforcement/collection agencies). Some government mandated rules have to be followed by BMI and ASCAP, but they are mainly concerned with percentage rates and other technical details. Which is why you see these old artists who were screwed out of royalties always sue the publishers.
All of which is to say, as the creator of an artistic work, you are the copyright holder, and there's no reason a BMI or ASCAP-type enforcement agency couldn't be created for podcasters. It's unlikely to happen for a lot of reasons, but theoretically it could. The question then becomes, would these networks even bother with podcasts if they had to pay royalties? And the answer to that is quite likely 'no,' since they only grudgingly pay music royalties (when they do pay them).