Diseases still do stat damage, keeping with tradition. (I considered making them do no stat damage, and just kill you if you failed 3 checks in a row, but that was an unsuitable solution for several reasons.)
The more interesting part is that each disease has a non-numeric effect associated with it. Sleeping sickness requires you to get more sleep each night, brain worms cause hallucinations in the hopes that they can trick you to spreading your awful affliction. (Brain worms are my favorite disease, although there are a few good ones.)
The most interesting part is that diseases are not (entirely) microbial, but are instead caused by evil spirits and their infectious miasmas. This has two big implications. First, it means that diseases can be placated with offerings. Second, it means that diseases can be driven out with exorcisms.
Credit Where Credit Is Due
Thank you +Thaumiel Nerub for speculating about diseases on G+ yesterday, which inspired me to start writing out some disease rules, and a double thank you to +Joseph Manola, whose criminally underappreciated blog has been quietly recording the proper way to do shamanic healing for some time now. (I honestly can't imagine how this post has no +1s.)