For one: crappy audio. None of the current DSLR cams allow quality sound recording, so if you are aiming to do something with reasonable quality you'll need to work "dual-system" -- meaning you need a separate sound recorder, and you have to somehow synchronize it with your video.
For two: there are some technical issues. Filmmaker magazine has a nice mini-article on the phenomenon, and it's the first time I've seen some of those technical concerns made clear.
Filmmakers disclose how they are shooting movies with still cameras.
“I was very afraid about the sensitivity of the camera to movement. I’d read a lot about how the rolling shutter in the Canon 5D can sometimes give a jelly effect. If you look at the focus push at the second mark, (26 seconds into the trailer at http://searchingforsonny.com) you’ll see what a lot of people are having problems with. We stayed away from handheld shots, more as a stylistic choice. We did a test before the shoot with handheld, some parts were a little too shaky. I think it’ll be a new camera technique to master. On the dolly, we used sandbags to weigh down the tripod. But even with a nice dolly with good track, we had to rehearse the shot over and over again. Every little bump could be seen on camera."
Tom Quinn had similar concerns. "These cameras are really meant to be operated on a tripod — the handheld has a few issues. For one, there is no optical image stabilizer for the video, so the small vibrations that a video camera would neutralize are present. Also the CMOS sensor creates a slight waver to the image on fast horizontal movement when shooting telephoto. For these two reasons we’ve been using a mix of monopods and tripods.”