Strain engineering on the thermal conductivity of graphene has attracted significant interest both from the fundamental physics and with applications in mind such as flexible electronics and sensors. But existing computational studies predict inconsistent impacts of strain on thermal transport in graphene, and it is challenging to experimentally quantify. Here, we measure the strain-dependent thermal conductivity of polycrystalline multilayer graphene (MLG) grown by chemical vapor deposition (CVD). The room-temperature thermal conductivity of the MLG film dramatically decreases from 551 +/- 28 W/m-K to 395 +/- 20 W/m-K with 1% uniaxial strain. The significant decrease (~28%) of the thermal conductivity is attributed to the increased phonon-grain boundary scattering rate and the reduced probability of specular phonon transmission across the grain boundaries. These findings demonstrate dynamically tunable thermal transport in graphene by strain engineering and emphasize that strain-dependent property characterization is crucial for ensuring the performance and reliability of flexible graphene-based electronics and sensors.