The amplitude ("Higgs") mode is a ubiquitous collective excitation related to spontaneous breaking of a continuous symmetry. We combine quantum Monte Carlo (QMC) simulations with stochastic analytic continuation to investigate the dynamics of the amplitude mode in a three-dimensional dimerized quantum spin system. We characterize this mode by calculating the spin and dimer spectral functions on both sides of the quantum critical point, finding that both the energies and the intrinsic widths of the excitations satisfy field-theoretical scaling predictions. While the line width of the spin response is close to that observed in neutron scattering experiments on TlCuCl_{3}, the dimer response is significantly broader. Our results demonstrate that highly nontrivial dynamical properties are accessible by modern QMC and analytic continuation methods.