We present a study of hydrogen, helium, and carbon millimeter-wave radio-recombination lines (RRLs) toward 10 representative positions throughout the Orion Nebula complex, using the Yebes 40 m telescope in the Q band (31.3 GHz to 50.6 GHz) at an angular resolution of about 45″ (~0.09 pc). The observed positions include the Orion Nebula (M42) with the Orion Molecular Core 1, M43, and the Orion Molecular Core 3 bordering on NGC 1973, 1975, and 1977. While hydrogen and helium RRLs arise in the ionized gas surrounding the massive stars in the Orion Nebula complex, carbon RRLs stem from the neutral gas of the adjacent photo-dissociation regions (PDRs). The high velocity resolution (0.3 km s−1) enables us to discern the detailed dynamics of the RRL emitting neutral and ionized gas. We compare the carbon RRLs with SOFIA/upGREAT observations of the [C II] 158 µm line and IRAM 30 m observations of the 13CO (J = 2−1) line (the complete map is presented here for the first time). We observe small differences in peak velocities between the different tracers, which cannot always be attributed to geometry but potentially to shear motions. Using the far-infrared [C II] and [13C II] intensities with the carbon RRL intensities, we can infer physical conditions (electron temperature Te and electron density ne, converted to hydrogen nuclei density nH by dividing by the carbon gas-phase abundance 𝒜c ≃ 1.4 × 10−4) in the PDR gas using nonlocal thermal equilibrium excitation models. For positions in OMC1, we infer ne ≃ 20–40 cm−3 and Te ≃ 210–240 K. On the border between OMC1 and M43, we observe two gas components with ne ≃ 2 cm−3 and ne ≃ 8 cm−3, and Te ≃ 100 K and Te ≃ 150 K. In M43, we infer ne ≃ 2–3 cm−3 and Te ≃ 140 K. The Extended Orion Nebula southeast of OMC1 is characterized by ne ≃ 2 cm−3 and Te ≃ 180 K, while OMC3 has ne ≃ 1 cm−3 and Te ≃ 130 K. Our observations are sensitive enough to detect faint lines toward two positions in OMC1, in the BN/KL PDR and the PDR close to the Trapezium stars, that may be attributed to RRLs of C+ or O+. In general, the RRL line widths of both the ionized and neutral gas, as well as the [C II] and 13CO line widths, are broader than thermal, indicating significant turbulence in the interstellar medium, which transitions from super-Alfvénic and subsonic in the ionized gas to sub-Alfvénic and supersonic in the molecular gas. At the scales probed by our observations, the turbulent pressure dominates the pressure balance in the neutral and molecular gas, while in the ionized gas the turbulent pressure is much smaller than the thermal pressure.