Graphene can at present be grown at large quantities only by the chemical vapor deposition method, which produces polycrystalline samples. Here, we describe a method for constructing realistic polycrystalline graphene samples for atomistic simulations, and apply it for studying their mechanical properties. We show that cracks initiate at points where grain boundaries meet and then propagate through grains predominantly in zigzag or armchair directions, in agreement with recent experimental work. Contrary to earlier theoretical predictions, we observe normally distributed intrinsic strength (∼50% of that of the monocrystalline graphene) and failure strain which do not depend on the misorientation angles between the grains. Extrapolating for grain sizes above 15 nm results in a failure strain of ∼0.09 and a Young's modulus of ∼600 GPa. The decreased strength can be adequately explained with a conventional continuum model when the grain boundary meeting points are identified as Griffith cracks.