Calling all Lighting Technicians: This is a True Challenge

http://i49.tinypic.com/2ir56vk.jpg

I am attempting to do a stop-motion film using a grid of bracelet boxes. As you can see, they have a curve to them, and pretty much reflect any light, anywhere in the room. I'm going to get a polarized lens, but I also need to figure out this lighting situation. As you can tell, those are jsut the fluorescents in the room. I haven't ordered the lights yet.

Anyone have any suggestions or helpful tips?

Polariser probably won't help much, as the light on the boxes won't be polarised sunlight.
Turn off the room fluorescents, and use bounced flash on the ceiling so that you get a large, even light source - as close to a cloudy sky as you could indoors.

Paul is right. I think you would need to light up the entire room from a source that cannot be "seen" by the tiles. flash swiveled backwards to bounce off the rear wall or a couple of flashes bouncing off both rear corners of the room should do well. turn off all other lights inside.

I could be way off base here, but I'm pretty sure bouncing flashes off the back wall is going to give you the same specular highlights you see in the center of this shot. This could be pulled off in a couple different ways with studio lights, but it might help if Mr Smith could tell us what equipment he has available.