We report on IRAM-30m/EMIR observations of 38 Herschel sources chosen as the brightest red sub-millimeter (sub-mm) sources in 18 Planck-selected fields drawn from the Planck high-z (PHz) protocluster candidates sample. These fields host overdensities of red Herschel sources, with high star formation rates (∼10,000 M yr −1 ) as derived from Planck measurements. The goals of these observations are to measure spectroscopic redshifts of the Herschel sources in the PHz fields, investigate the origin of their bright sub-mm emission, and find evidence of their association with high-z protoclusters. We detect 40 CO lines on a total of 24 bright (S 350µm >40 mJy) Herschel sources in 14 of the 18 PHz fields. The measured average redshift is = 2.25±0.09, spanning a range from 1.32 to 2.75. We measure redshifts for multiple Herschel sources in close projected proximity in eight PHz fields. In half of those fields we detect from two to three objects at similar redshifts, supporting the idea that the PHz fields contain high-z protoclusters. The detection of sources at different redshifts in the same field demonstrates that foreground and background sources also contribute to the total sub-mm emission. We compare the properties of the molecular gas, and of the star formation activity of our sources with samples of normal star-forming galaxies (SFGs), sub-mm galaxies (SMGs), and CO-detected cluster and protocluster galaxies drawn from the literature at similar redshifts. We find that the PHz-IRAM sources are mainly normal SFGs, with only ∼20% undergoing a starburst phase. The PHz-IRAM sources are characterized by star formation rates ( = 1043±157 M yr −1 ) and gas masses ( = (4.0±0.7)×10 11 M ) that are, on average, eight and five times higher than those typical of normal SFGs at similar redshifts. Their dust temperatures ( = 29.2±0.9 K), and depletion timescales (<τ dep >=0.47±0.07 Gyr) are instead consistent with those of normal SFGs. The analysis of the CO spectral line energy distribution, available for ten PHz-IRAM sources, peaks at a low quantum rotation number (J up = 3) in most of the cases, implying low gas excitation. These properties imply that a significant fraction of the PHz-IRAM sources contains extended, and cold molecular gas reservoirs at low excitation, and that their star-formation is driven by secular processes. Multiplicity and moderate gravitational lensing might also play a role in producing the observed properties. Some of these properties are also observed in other CO-detected infrared-luminous protocluster galaxies at z 1.3-3. We find that the highest star-forming protoclusters drawn from the largest volume simulations available to date, have similar SFRs as the PHz protoclusters, but separate out into a larger number of star-forming galaxies. Millimeter and CO observations at higher spatial resolution than those presented here would be necessary to further elucidate the properties of our PHz-IRAM sources, and determine which mechanisms drive star-formation in infrared-lum...