This makes a lot more sense with either a fixed-gear bike, or an old 10- or 12-speed on which the teeth of the sprockets and chainrings weren't designed to facilitate better shifting. Here are the issues:
- deraillers - to my knowledge, no one makes deraillers that operate in the reverse direction. You could contract Paul to build you reverse deraillers or something, but this would be crazily expensive. This basically forces you into the fixed-gear or singlespeed world, or at least the non-derailler world, if you want to run your chain on the left side of the bike.
- chainring and sprocket teeth - modern chainrings and sprocket teeth are ramped and stuff to aid shifting. Running stuff in the reverse would render these ramps and pins ineffective, and perhaps counterproductive. You could do fine with older gears where the teeth aren't twisted, and there's no ramps in the cogs and pins in the chainrings.
- freewheel or cassette rotation direction: the cassette or freewheel contains rachets pointed in one direction, that allows you to coast and pedal backwards, but engages the pedals when you pedal. I suspect someone manufactures freehub bodies that coast in the opposite direction as normal hubs. But maybe not - there'd be no point in doing this unless you could get opposite-side deraillers.
- threading of the rear hub for freewheel or track sprocket - there may well be track hubs with reverse threading out there, although I've never heard of them. A regular track hub with track cog and well-tightened lockring might work here, but I'm not sure - they're designed to have the cog on tight for resisting the pedals (braking of a sort) but I'm not sure if the design is up to having continuous stress in the opposite direction as design intended.
Sheldon Brown has a bike with chain on left side. But it's a funky old 3-speed fixed-gear hub, that'll let you do this. Click on the picture for more info...
edit: juicemouse beat me to it.