In my geographical area the owners deal with this (assuming they don't choose to pay up and become full-fledged on-the-up-and-up music venues) by insisting on "original singer-songwriter material only."
Apparently the reasoning is that it is much easier to defend ownership of a song one has written than to research (not to mention prove) non-ownership of a traditional song -- especially considering how many popularly thought of as trad songs are owned by someone.
The result is the same for a singer of traditional folk. If it isn't yours, you have a devil of a time finding somewhere to sing it.
-Glenn