The metal-organic halide perovskites may have excellent photovoltaic properties, useful to create cheap and efficient solar cells, photodetectors and LEDs with a wide range of colors. Great efforts are spent for improving their scarce stability and clarifying the mechanisms that allow the photocarriers to have so long lifetimes. These phenomena are closely connected to the structural transformations and to the dynamics of the reorienting organic cations, both of which can be studied by anelastic measurements, here combined with dielectric spectroscopy. The complex dynamic Young’s modulus and permittivity of MAPI and FAPI have been measured on samples obtained by pressing the powders into bars or disks. Both perovskites are cubic at and above room temperature (α phase), with freely rotating FA and MA cations, and undergo two tilt transitions of the PbI₆ octahedra into a tetragonal (β) and orthorhombic (γ) phase, accompanied by losses of the orientational degrees of freedom of the MA and FA molecular cations. These α → β and β → γ transitions are accompanied respectively by sharp softening and stiffening of the Young’s modulus, while the permittivity has a major drop from a paraelectric rise at the β → γ transition of MAPI and is practically flat everywhere in FAPI. From these behaviors we argue that the α∕β transition is mainly due to octahedral tilting while the latter to the blocking of MA/FA reorientational modes linearly coupled with strain. Comparisons are made with elastic anomalies found in other hybrid 3D and layered perovskites.