We have performed long time scale molecular dynamics simulations of the cubic and tetragonal phases of the solid lithium-ion electrolyte Li7La3Zr2O12 (LLZO), using a first-principles parametrized interatomic potential. Collective lithium transport was analyzed by identifying dynamical excitations: persistent ion displacements over distances comparable to the separation between lithium sites, and stringlike clusters of ions that undergo cooperative motion. We find that dynamical excitations in c-LLZO (cubic) are frequent, with participating lithium numbers following an exponential distribution, mirroring the dynamics of fragile glasses. In contrast, excitations in t-LLZO (tetragonal) are both temporally and spatially sparse, consisting preferentially of highly concerted lithium motion around closed loops. This qualitative difference is explained as a consequence of lithium ordering in t-LLZO and provides a mechanistic basis for the much lower ionic conductivity of t-LLZO compared to c-LLZO.