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The effect of the Berry phase is included explicitly in the wavefunction describing conduction electrons in a crystal composed of periodically arrayed Jahn-Teller centers that have conically intersecting potential energy surfaces. The Berry phase can make a drastic change in the band structure, leading generally to the formation of an orbital-density-wave state. We discuss implications of our theory and possible relations to the orbital ordering observed in the manganese perovskites., Comment: 4 pages. 3 eps figures. RevTex. Submitted to Phys. Rev. Lett