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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 $K_{s}=7.18$ magnitude G8 V dwarf at a distance of only 54 parsecs. We validate the planetary nature of the transit signals by combining five 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 ($R_{b} = 3.42^{+0.11}_{-0.11}\,R_{\oplus}$, $R_{c}=3.23_{-0.11}^{+0.11}\,R_{\oplus}$, and $R_{d}=3.16_{-0.11}^{+0.11}\,R_{\oplus}$) and their orbits are consistent with a stable, circular, and co-planar architecture near mean motion resonances of 1:3 and 3:4 ($P_{b}=8.88$ days, $P_{c}=28.58$ days, and $P_{d}=38.35$ 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 multi-wavelength transmission spectroscopy of all three planets' atmospheres, HD 191939 can offer valuable insight into multiple sub-Neptunes born from a proto-planetary disk that may have resembled that of the early Sun.