We have recently developed a novel molecular-dynamics simulation method to grow polycrystals from a melt containing randomly oriented crystalline seeds. The resulting microstructures contain only randomly oriented (i.e., high-energy) grain boundaries. We find that these grain boundaries, which are highly constrained by their close proximity to grain junctions, are highly disordered in fcc metals and amorphous in silicon. From simulations of infinitely extended high-energy grain boundaries in bicrystals, we find that such highly disordered and amorphous grain boundaries are actually the thermodynamic ground state; by contrast, low-energy grain boundaries are crystalline. High-energy grain boundaries in diamond, however, are structurally ordered at the expense of a significant amount of graphite-like bonding. We show that these complex grain boundary structures have important effects on properties including grain boundary diffusion (fcc metals and silicon), grain boundary diffusion creep (silicon) and grain boundary electrical activity and strength (diamond). The implications for engineering materials with prescribed properties are discussed.