"Sheathing" is a construction term for builders. It refers to the "skin" or layer of material placed over the 2x stud framing of walls and roofs. Typically (depending on building codes in your area) it can refer to 3/4 plywood, particle board, OSB or other sheets of solid material (4'x8' or larger). It is generally "rough" in texture as it is meant to be covered up by siding, insulation, Tyvek wrap etc, etc.

When it is placed over floor joists it is called "sub-flooring" but is essentially referring to the same thing. It is likewise rough because it will be covered up with other material.

Smooth-sided plywood in comparison is much more expensive. One-sided plywood (smooth on one side) is reasonably cheaper, but I propose another solution:

Insteand of building platforms (which can be costly and noisy) you may want to consider laying plywood on top of some kind of soft underlayment (rubber or thin foam) that is used in the installation of hardwood flooring.

Covering your cement with heavy mil plastic will act as a water barrier (in case there's a danger of seepage). A layer of thin foam is placed on top of that and then the plywood.

For a much softer and sturdier floor, you could "sandwich" another layer of foam between two layers of sheathing (or subfloor).

On top of the plastic/foam combo I previously mentioned, lay exterior quality 1/2 plywood, OSB, or particle board (it's cheaper).   You could even use tongue-and-groove material, but that is more difficult to work with.

Then a second layer of foam on top of that followed by one-sided 3/4 ply in an alternate pattern (smooth side up, of course), screwing it down firmly into the 1/2 inch material below (NOT the concrete!), staggering and overlapping the seams.

If don't recommend doing the ENTIRE floor (of a 5000 sq ft warehouse) because that would be cost-prohibitive, but merely map out a section of floor that replicates the size of the performance space.

Frame the perimeter with green-treated 2x4's bolted and glued directly into the concrete below. The plywood flooring would then be layed within this framework and "float" (meaning it is unattached to the concrete or 2x4 frame).   Making a sandwich of material would ensure the plywood would remain in place, and not curl or shift at the seams.

Good luck!

I still haven't really decided what to do at this point but I've gone ahead and started framing out some 4 x 8 platforms because either way we're going to need them. Money's so tight right now (just dropped the cash on the venue) that I'll probably just follow junglejim's lead and go with foam sandwiched in 1/2" ply.

Any specifics on the foam?

On a tangent could one cover the top of the foam with 1/4" luan. I get a pretty good deal on those as I buy a lot for the flats. But not only would it be cheaper but it would provide a paint-able surface.  Too thin?