The position of the RIAA is a lot trickier than you'd think: you don't actually own a copy of the music. What you buy is a license to listen to it on their terms. The copyright, or the right to reproduce the material (on whatever terms), always remains with them.
Nick, it is both a criminal and a civil matter.
Personally, I don't believe copyright law serves any purpose, and I think that every argument about "junk lawsuits" and "bogus civil claims" applies to modern copyright practice. It's an invented class of property, and it was originally sold by the same people who used government monopolies to promote the exploitation of colonial resources.
Morally, I think the only legitimate purpose of copyright is to promote development. If some law can't, in hard numbers, demonstrably improve development of new works, it should be discarded.
Legally, however, the current state is pretty clear on this point. It's illegal and tortious (with junk penalties attached to promote junk lawsuits) to replace music files from an unauthorised source.