Monolayer transition metal dichalcogenides (TMDs) are known to be highly sensitive to externally applied tensile or compressive strain. In particular, strain can be exploited as a tool to control the optical response of TMDs. However, the role of excitonic effects under strain has not been fully understood yet. Utilizing the strain-induced modification of electron and phonon dispersion obtained by first principle calculations, we present in this work microscopic insights into the strain-dependent optical response of various TMD materials. We show that the different changes in the excitonic linewidth of diverse TMD monolayers are due to the strain-induced modification of the relative spectral position of bright and dark excitonic states. Our theoretical results explain well the observed partially opposite changes in the excitonic linewidth of different TMDs at room temperature. Furthermore, we predict the linewidth behavior of excitonic resonances in strained TMDs for tensile and compressive strain at low temperatures.