Data on the inclusive production of the neutral vector mesonsρ0(770),ω(782), K*0(892), andφ(1020) in hadronic Z decays recorded with the ALEPH detector at LEP are presented and compared to Monte Carlo model predictions. Bose-Einstein effects are found to be important in extracting a reliable value for theρ0 production rate. An averageρ0 multiplicity of 1.45±0.21 per event is obtained. Theω is detected via its three pion decay modeω→π+π−π0 and has a total rate of 1.07±0.14 per event. The multiplicity of the K*0 is 0.83±0.09, whilst that of theφ is 0.122±0.009, both measured using their charged decay modes. The measurements provide information on the relative production rates of vector and pseudoscalar mesons, as well as on the relative probabilities for the production of hadrons containing u, d, and s quarks.