As written by Justin you need to check the normals first. Usually going into TEXTURED mode in the 3d view will let you see if you have a normals problem. In which case select the mesh and go into edit mode (depending on your version activating normals visualization requires different steps), and swap them as needed.
**On the other hand, if it's not a normals issue**, be sure that you don't have a scaling problem in the blender editor: select the mesh and check its scale (properties panel => object tab=> scale). If you have negative values there, normals will be messed up regardless of their correct displaying in blender.
**TO FIX:** select the mesh in object mode and press CTRL A (and then apply scale).
The scale will be reverted to 1 1 1 and thus the model will be correctly exported (PRESERVING ANIMATIONS, nonetheless).
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