I was wondering about the genitive case here as well. I found an explanation here: http://blogs.transparent.com/russian/accusative-and-genitive-in-negative-russian-sentences/
I think the ending sums up my feelings nicely:
"If you feel confused, take solace in the fact that many native speakers tend to be just as confused about the use of accusative and genitive cases in negative sentences. Remember, the more you speak, the higher the chances you will get it right!"
I've done a small bit of research on the topic of negation in Russian, and it seems to be a really difficult topic, when you get away from the obvious нет [genitive]. I like Peatsickle's explanation about direct objects of negated verbs being cast in genitive, but I'm not sure if that can be a "rule". It seems more complicated than that.
Similar phenomena occur in other European languages as well, even when there are no cases. For example, in Italian you could say "Non so di una risposta" instead of "Non so la risposta". It sounds like "I don't know of a possible answer" rather than "I don't know the answer".