β-decay properties of even-even and odd-A neutron-rich Ge, Se, Kr, Sr, Zr, Mo, Ru, and Pd isotopes involved in the astrophysical rapid neutron capture process are studied within a microscopic proton-neutron quasiparticle random-phase approximation. The underlying mean field is based on a self-consistent Skyrme Hartree-Fock + BCS calculation that includes deformation as a key ingredient. The isotopic evolution of the various nuclear equilibrium shapes and the corresponding charge radii are investigated in all the isotopic chains. The energy distributions of the Gamow-Teller strength, as well as the β-decay half-lives are discussed and compared with the available experimental information. It is shown that nuclear deformation plays a significant role in the description of the decay properties in this mass region. Reliable predictions of the strength distributions are essential to evaluate decay rates in astrophysical scenarios.