We report the results of an ultrafast, direct structural measurement of optically pumped phonons in a Cr thin film using ultrashort X-ray pulses from a free-electron laser. In addition to measuring and confirming the known long-wavelength dispersion relation of Cr along a particular acoustic branch, we are able to determine the relative phase of the phonons as they are generated. The Cr sample exhibits two generation mechanisms for the phonons: the releasing of a preexisting charge density wave at higher frequencies, and the creation of an acoustic strain pulse via laser heating that dominates at lower frequencies. For the latter mechanism, we are able to measure the frequency dependence of the time required to generate the phonons. To explain the observed magnitude and slope of the delays, we perform first-principles simulations in the framework of density functional perturbation theory and ab initio molecular dynamics to fit anharmonic phonon models. These results show that the wave-vector dependence of the electron-phonon coupling is the driving mechanism behind the delay times: Phase-space limitation leads to higher times near the zone center. The absolute magnitudes of the delay times measured are found to be much shorter than the equilibrium electron-phonon coupling times we compute, indicating that the coupling strength is greatly enhanced when the electronic system is out of equilibrium with the lattice, as has been seen in bismuth and other systems.