Glitches (spurious transitions) are common in electronic circuits. In this paper we present a novel approach to reduce dynamic power in FPGAs by reducing glitches during the routing step. This approach involves finding alternative routes for early-arriving signals, so that signal arrival times at LUTs are aligned and no glitches are generated. This approach does not require additional circuitry to balance signals as done in previous work, but uses the available programmable routing resources instead. We develop an efficient algorithm to find routes with target delays. Based on this algorithm, we then build a glitch-aware router, named GlitchReroute, aiming at reducing dynamic power. To the best of our knowledge, this is the first glitch-aware routing algorithm for FPGAs. Experiments show that an average of 23% reduction in glitch power is achieved, which translates into a 9.8% reduction in dynamic power, compared to the glitch-unaware VPR router.