The inclusive production cross sections of the strange vector mesons K * 0 , K * 0 , and φ have been measured in interactions of 920 GeV protons with C, Ti, and W targets with the HERA-B detector at the HERA storage ring. Differential cross sections as a function of rapidity and transverse momentum have been measured in the central rapidity region and for transverse momenta up to pT = 3.5 GeV/c. The atomic number dependence is parametrised as σpA = σpN * A α , where σpN is the proton-nucleon cross section. Within the phase space accessible, α(K * 0) = 0.86 ± 0.03, α(K * 0) = 0.87 ± 0.03, and α(φ) = 0.96 ± 0.02. The total proton-nucleon cross sections, determined by extrapolating the differential measurements to full phase space, are σ pN→K * 0 = (5.06 ± 0.54) mb, σ pN→K * 0 = (4.02 ± 0.45) mb, and σ pN→φ = (1.17 ± 0.11) mb. For all resonances the Cronin effect is observed; compared to the measurements of Cronin et al. for K ± mesons, the measured values of α for φ mesons coincide with those of K + mesons for all transverse momenta, while the enhancement for K * 0 /K * 0 mesons is smaller.