I figured out the problem. All the objects have textures assigned to them, but the associated images don't know where to go because the objects aren't UV mapped to them. If you unwrap everything, it should work fine, but an interrim solution is to either unlink the image files from the texture (in which case everything should be white), or to do a quick-and-dirty UV mapping (even as simple as U > Reset) just so it has somewhere to put the image around your model (in which case it should look jumbled and weird).
(On a tangentially-related note, make sure "map input" is set to UV instead of Orco.)