We review recent studies of spin dynamics in rare-earth orthorhombic perovskite oxides of the type RMO3, where R is a rare-earth ion and M is a transition-metal ion, using single-crystal inelastic neutron scattering (INS). After a short introduction to the magnetic INS technique in general, the results of INS experiments on both transition-metal and rare-earth subsystems for four selected compounds (YbFeO3, TmFeO3, YFeO3, YbAlO3) are presented. We show that the spectrum of magnetic excitations consists of two types of collective modes that are well separated in energy: gapped magnons with a typical bandwidth of <70 meV, associated with the antiferromagnetically (AFM) ordered transition-metal subsystem, and AFM fluctuations of <5 meV within the rare-earth subsystem, with no hybridization of those modes. We discuss the high-energy conventional magnon excitations of the 3d subsystem only briefly, and focus in more detail on the spectacular dynamics of the rare-earth sublattice in these materials. We observe that the nature of the ground state and the low-energy excitation strongly depends on the identity of the rare-earth ion. In the case of non-Kramers ions, the low-symmetry crystal field completely eliminates the degeneracy of the multiplet state, creating a rich magnetic field-temperature phase diagram. In the case of Kramers ions, the resulting ground state is at least a doublet, which can be viewed as an effective quantum spin-1/2. Equally important is the fact that in Yb-based materials the nearest-neighbor exchange interaction dominates in one direction, despite the three-dimensional nature of the orthoperovskite crystal structure. The observation of a fractional spinon continuum and quantum criticality in YbAlO3 demonstrates that Kramers rare-earth based magnets can provide realizations of various aspects of quantum low-dimensional physics.