Existing examples of Peierls-type 1D systems on surfaces involve depositing metallic overlayers on semiconducting substrates, in particular, at step edges. Here we propose a new class of Peierls system on the (101[over ¯]0) surface of metal-anion wurtzite semiconductors. When the anions are bonded to hydrogen or lithium atoms, we obtain rows of threefold coordinated metal atoms that act as one-atom-wide metallic structures. First-principles calculations show that the surface is metallic, and below a certain critical temperature the surface will condense to a semiconducting state. The idea of surface scaffolding is introduced in which the rows are constrained to move along simple up-down and/or sideways displacements, mirroring the paradigm envisioned in Peierls's description. We predict that this type of insulating state should be visible in the partially hydrogenated (101[over ¯]0) surface of many wurtzite compounds.