Parts produced via laser powder-bed fusion additive manufacturing exhibit complex microstructures that depend on processing variables and often vary widely in crystallographic texture and grain morphology. The need to understand, predict, and control these microstructural variations motivates the development of modeling tools capable of accurately predicting LPBF microstructures. Monte Carlo (MC) Potts models have been employed to successfully model the formation of grain structures in additively manufactured parts but have lacked the ability to simulate crystallographic texture. We present an extension of the MC Potts model that assigns an orientation to each grain and penalizes growth of solid into the fusion zone based on proximity of the nearest 〈100〉 crystal direction to the local temperature gradient direction. This allows for crystallographically selective growth to drive texture formation during the development of the solidification microstructure in each melt track. LPBF builds of alloy 718 with a unidirectional scan pattern provided microstructures with substantial variations in grain size, grain morphology, and texture. These distinctive albeit atypical microstructures were used to validate the simulation method, i.e. good agreement was obtained between the simulated and experimental grain shapes and textures.