I just started playing on Linux and it has the same problems - I've tracked it down to incompatible shaders. But unless the mod creators are willing to make a fix just for mac and linux users since they work in Windows, there's not much you can do. I downloaded openbrf and started messing with the resource files, but I'm basically taking shots in the dark as I have very little understanding of Warband modding - by changing some shaders in armor textures that I've noticed to be broken on linux, I've managed to fix at least some issues, but it's not really practical for most mac/linux users.

If you are still reading this and still have issues, check the comments page - Mindeveler posted a link that you can follow to find a fix.