We present [O_ III] 88μm observations towards four Herschel-selected dusty star-forming galaxies (DSFGs; log_10 μL_ IR/L_⊙ = 13.5 - 14 at z = 2.9 - 4) using the Atacama Compact Array (ACA) in Bands 9 and 10. We detect [O_ III] emission in all four targets at >3σ, finding line luminosity ratios (L_ [O_ III] / L_ IR = 10^-4.2 to 10^-3) similar to local spiral galaxies, and an order of magnitude lower when compared with local dwarf galaxies as well as high-redshift Lyman-break galaxies. Using the short-wavelength capabilities of the ACA, these observations bridge the populations of galaxies with [O_ III] emission at low redshift from space missions and at high redshift from ground-based studies. The difference in [O_ III] emission between these DSFGs and other high-redshift galaxies reflects their more evolved stellar populations (> 10 Myr), larger dust reservoirs (M_ dust ∼ 10^9 - 11 M_⊙), metal-rich interstellar medium (Z ∼ 0.5 - 2 Z_⊙), and likely weaker ionization radiation fields. Ancillary [C_ II] emission on two targets provide L_[ O_ III] / L_[ C_ II] ratios at 0.3 - 0.9, suggesting that ionized gas represents a smaller fraction of the total gas reservoir in DSFGs, consistent with theoretical models of DSFGs as transitional systems between gas-rich, turbulent disks and more evolved, gas-poor galaxies. Expanding samples of DSFGs with [O_ III] emission will be key to place this heterogeneous, poorly-understood galactic phase in its astrophysical context.