Diffractive photoproduction of vector mesons, γp → VY, where Y is a proton-dissociative system, has been measured in e+p interactions with the ZEUS detector at HERA using an integrated luminosity of 25 pb-1. The differential cross section, dσ/dt, is presented for 1.2 < -t < 12 GeV2, where t is the square of the four-momentum transferred to the vector meson. The data span the range in photon-proton centre-of-mass energy, W, from 80 GeV to 120 GeV. The t distributions are well fit by a power law, dσ/dt ∝ (-t)-n. The slope of the effective Pomeron trajectory, measured from the W dependence of the ρ0 and φ cross sections in bins of t, is consistent with zero. The ratios dσγp→φY/dt to dσγp→ρ0Y/dt and dσγp→J/ψY/dt to dσγp→ρ0Y/dt increase with increasing -t. Decay-angle analyses for ρ0, φ and J/ψ mesons have been carried out. For the ρ0 and φ mesons, contributions from single and double helicity flip are observed. The results are compared to expectations of theoretical models.