You're going to have to synch all three cameras when you 're doing post. If you're able to view the audio as waveforms, synching is just a matter of matching the waveforms. If not, listening to the audio is much easier than trying to synch using visual cues. Just listen for the echo effect and move the audio one way or the other, you'll easily be able to tell if you're getting closer or futher off. Another useful trick is to focus all three cameras on a clapboard, then clap it. In any case, DO NOT STOP ANY CAMERA UNTIL YOU'RE FINISHED. Otherwise you'll be spending all your time resynching your video channels.
If you decide to run the mix out of your audio mixer, make sure it provides the ability to send a "mic out" signal. You won't be able to get decent, or even acceptable, sound from the "line out" signal. The biggest advantage of using a mixer is that you can have someone dedicated to monitoring & riding the sound. By having them adjusting audio levels as you talent speaks generally makes for a better audio track. And you only need to record your primary audio on one camera, you can use the on camera mics to synch with, then mute them for the final project. I would not recommend that you run the mics to a camera, then run camera sound to the mixer and then record the mix. It greatly decreases you ability to adjust the sound during the recording. If for some reason you are unable to record your audio on one of the cameras, DO NOT USE A TAPE RECORDER. You must record on a digital recording device, a DAT, a mini-disc, a CD or directly into a computer. These options do allow you to use the line out signal from your audio mixer.
I wish you luck with your production. I remember back to when I first got started in video production, of course it was all on linear tape back then. But multi-cam recording can really improve the look & feel of your productions. Putting up all with all the pre-production hassles will definitely pay off in the long run. And by the way, have fun while you work, it makes the show look even better.
Tony Clark, Barefoot Media Productions