First-principles calculations of flexoelectricity, the electromechanical coupling whereby a strain gradient produces a macroscopic polarization, are challenging due to the inherent breakdown of crystal periodicity. An effective workaround to these difficulties was demonstrated in the recent past via a careful long-wavelength analysis of acoustic phonons. Following these guidelines, the authors develop here a practical formalism and a code implementation providing the lattice-mediated contributions to the bulk flexoelectric tensor. This work completes the first-principles theory and implementation of bulk flexoelectricity and makes it available for public use.