We have performed Monte Carlo (MC) simulations on two-dimensional systems of quadrupole particles confined to a triangular lattice in order to determine the conditions that permit the formation of a limit-periodic phase. We have found that limit-periodic structures form only when the rotations of the particles are confined to a set of six orientations aligned with the lattice directions. Related structures including striped and unidirectional rattler phases form when π/π66 rotations or continuous rotations are allowed. Order parameters signaling the formation of the limit-periodic structure and related structures are measured as a function of temperature. Our findings on the formation of the limit-periodic structure elucidate features relevant to the experimental creation of such a structure, which is expected to have interesting vibrational and electromagnetic modes.