Three-dimensional hybrid organic–inorganic lead halide perovskites are promising photovoltaic and light-emitting materials. A key phenomenon relevant for their optoelectronic applications is electron–phonon coupling. Since it can be strongly modified by structural deformation and changes in the dynamics of molecular cations, it is of great importance to study the temperature dependence of phonon properties of hybrid perovskites. In this work, temperature-dependent Raman scattering studies of FAPbBr3 and MAPbBr3 single crystals are reported in the 1800–22 cm−1 and 300–90 K ranges. The Raman data of MAPbBr3 showed clear anomalies near 236, 155 and 148 K, which were attributed to Pm3¯m→I4/mcm→P4/mmm (or Imma)→Pnma phase transitions. They also provided strong evidence that crystal structure of the phase stable in the 155–148 K range is very similar to structure of the I4/mcm phase, not structure of the lowest-temperature Pmna phase, as suggested in some reports. Therefore, the symmetry of this phase seems to be more likely P4/mmm rather than Imma. An analysis of the temperature evolution of MAPbBr3 Raman modes revealed that the phase transitions near 236 and 155 K are associated with weak distortion of the inorganic subnetwork and changes in the dynamics of MA+ ions. Very pronounced changes in the lattice modes region and a narrowing of bands below 148 K indicated that the phase transition to the Pnma phase is triggered by a freezing of MA+ motions, which in turn leads to strong distortion of the inorganic subnetwork. Raman studies of FAPbBr3 showed that this material behaves in a very different way than MAPbBr3. First of all, the molecular dynamics of FA+ cations are not frozen even in the lowest-temperature Pnma phase. Moreover, the distortion of the inorganic subnetwork is small in the Pnma phase. The observation of weak anomalies in the lattice modes region confirmed, however, that the two crystallographically resolved phase transitions (Pm3¯m→P4/mbm near 260 K and P4/mbm→Pnma near 150 K) lead to weak distortion of the inorganic subnetwork. On the other hand, an analysis of FA+ internal modes indicated that these transitions, as well as two other crystallographically unresolved transitions near 120 and 180 K, are triggered by a change of FA+ dynamics.