Film Riot's style of fight scene cinematography is really an ode to the current Hollywood style of fight scene. In my head I just think of "Bourne" cinematography. That said, the Film Riot guys are still learning (yes just like all the rest of us) and their fight scene cinematography wasn't perfect or brilliant by any means. It was effective enough to tell a story.
Camera movement is essential to this kind of style. But to me "camera shake" is a much heavier word. It feels like when people mention "camera shake," I think of the intense turbulence with the camera we were used to seeing in the early 2000s.
I think a stedicam approach is the best for a fight scene. Use camera shake during most intense hits, or use a camera whip to reveal someone or something. I was working with a very young and gifted cinematographer (it still blows my mind to this day how he grasped the concepts of Hollywood action cinematography at such a young age.) He would use the camera movement to "Reveal." He'd move the camera where it needed to be to reveal a attack (in the direction an attack was going,) to reveal an impact or damage (example: One character's punches another character's head. The attackers fist swings from the left side of the screen to right. Use camera to swing from the left, far to the right then back, on to the subject. Not shake the camera left to right, left to right, left to right, left to right. One forceful camera whip goes a longer way then just shaking the camera out of control. One method adds energy, one just looks amateur) or reveal an object/weapon used by a character. There is a lot of smooth but fast camera movement, but very few uses of actual camera "shake" or heavy turbulence.
If you think about it, when we look at things in real life, very few times do our eyes just stay dead focused on one thing. Even when we talk to our friends, our head and eyes move ever so slightly. I think that's the appeal of camera movement / camera shake.
A basic example of camera shake vs camera movement in terms of efficiency, is tracking with someone with or without a steadicam. If you track with someone who's running and you're running yourself, if it's handheld and without any steadicam or stability- it just looks like a bad shaky Blair witch shot. If you track someone with a steadicam and they're running, it adds energy and the audience can still follow them. Camera movement adds a live feel, camera shake disorients. You want to disorient them sometimes (so DO use camera shake) but don't over do it. You don't want them to look like they started fighting and then an earthquake happened.
Use lots of close ups and if you can go with a steadicam.