We present the discovery of three sub-Neptune-sized planets transiting the nearby and bright Sun-like star HD 191939 (TIC 269701147, TOI 1339), a Ks = 7.18 mag G8 V dwarf at a distance of only 54 pc. We validate the planetary nature of the transit signals by combining 5 months of data from the Transiting Exoplanet Survey Satellite with follow-up ground-based photometry, archival optical images, radial velocities, and high angular resolution observations. The three sub-Neptunes have similar radii ( ${R}_{{\rm{b}}}={3.42}_{-0.11}^{+0.11}$ , ${R}_{{\rm{c}}}={3.23}_{-0.11}^{+0.11}$ , and ${R}_{{\rm{d}}}={3.16}_{-0.11}^{+0.11}\,{R}_{\oplus }$ ), and their orbits are consistent with a stable, circular, and coplanar architecture near mean-motion resonances of 1:3 and 3:4 (Pb = 8.88, Pc = 28.58, and Pd = 38.35 days). The HD 191939 system is an excellent candidate for precise mass determinations of the planets with high-resolution spectroscopy due to the host star's brightness and low chromospheric activity. Moreover, the system's compact and near-resonant nature can provide an independent way to measure planetary masses via transit timing variations while also enabling dynamical and evolutionary studies. Finally, as a promising target for multiwavelength transmission spectroscopy of all three planets' atmospheres, HD 191939 can offer valuable insight into multiple sub-Neptunes born from a protoplanetary disk that may have resembled that of the early Sun.