Theoretically challenging, the understanding of the dynamical response in quantum antiferromagnets is of great interest, in particular for both inelastic neutron scattering (INS) and nuclear magnetic resonance (NMR) experiments. In such a context, we theoretically address this question for quasi-one-dimensional quantum magnets, e.g. weakly coupled spin chains for which many compounds are available in Nature. In this class of systems, the dimensional crossover between a three-dimensional ordered regime at low temperature towards one-dimensional physics at higher temperature is a non-trivial issue, notably difficult concerning dynamical properties. Here we present a comprehensive theoretical study based on both analytical calculations (bosonization + random phase and self-consistent harmonic approximations) and numerical simulations (quantum Monte Carlo + stochastic analytic continuation) which allows us to describe the full temperature crossover for the NMR relaxation rate $1/T_1$, from one-dimensional Tomonaga-Luttinger liquid physics to the three-dimensional ordered regime, as a function of inter-chain couplings. The dynamical structure factor, directly probing the INS intensity, is also computed in the different regimes.