A new peridynamic (PD) formulation is developed for cubic polycrystalline materials. The new approach can be a good alternative to traditional techniques such as finite element method (FEM) and boundary element method (BEM). The formulation is validated by considering a polycrystal subjected to tension-loading condition and comparing the displacement field obtained from both PDs and FEM. Both static and dynamic loading conditions for initially damaged and undamaged structures are considered and the results of plane stress and plane strain configurations are compared. Finally, the effect of grain boundary strength, grain size, fracture toughness, and grain orientation on time-to-failure, crack speed, fracture behavior, and fracture morphology are investigated and the expected transgranular and intergranular failure modes are successfully captured. To the best of the authors' knowledge, this is the first time that a PD material model for cubic crystals is given in detail.