Context. Abundances of iron-peak and alpha-elements are poorly known in Orion, and the available measurements yield contradictory results. Aims. We measure accurate and homogeneous elemental abundances of the Orion subgroups ONC and OB1b, and search for abundance differences across the Orion complex. Methods. We present FLAMES/UVES spectroscopic observations of 20 members of the ONC and OB1b. We measured radial velocity, veiling, effective temperature using two spectroscopic methods, and determined the chemical abundances of Fe, Na, Al, Si, Ca, Ti, and Ni using the code MOOG. We also performed a new consistent analysis of spectra previously analyzed by our group. Results. We find three new binaries in the ONC, two in OB1b, and three non-members in OB1b (two of them most likely being OB1a/25 Ori members). Veiling only affects one target in the ONC, and the effective temperatures derived using two spectroscopic techniques agree within the errors. The ONC and OB1b are characterized by a small scatter in iron abundance, with mean [Fe/H] values of -0.11+/-0.08 and -0.05+/-0.05, respectively. We find a small scatter in all the other elemental abundances. We confirm that P1455 is a metal-rich star in the ONC. Conclusions. We conclude that the Orion metallicity is not above the solar value. The OB1b group might be slightly more metal-rich than the ONC; on the other hand, the two subgroups have similar almost solar abundances of iron-peak and alpha-elements with a high degree of homogeneity.