We analyze the dynamic failure of concrete using two and three-dimensional finite-element models. In particular, the current results extend, using a unique parallel computing approach, previously published two-dimensional results. Concrete is treated at the mesoscale with an explicit representation of coarse aggregates and mortar paste. The propagation and coalescence of cracks are modeled with dynamically inserted cohesive elements. Stress-strain response, dissipated fracture energy and crack evolution are compared under tensile loading at several strain rates. The artifacts of the two-dimensional approach regarding microcrack coalescence are discussed in detail. Nonetheless, the dynamic increase factors for the peak strength and dissipated fracture energy show that in both two and three dimensions, micro-inertial effects are not enough to simulate the rate dependency in concrete with a simple rate-independent cohesive law. Rate-dependent damage parameters can be introduced to obtain a more accurate dynamic response.