The DL-Lite description logics allow for modeling domain knowledge on top of databases and for efficient reasoning. We focus on metric temporal extensions of DL-Lite_bool and its fragments, and study the complexity of satisfiability. In particular, we investigate the influence of rigid and interval-rigid symbols, which allow for modeling knowledge that remains valid over (some) time. We show that especially the latter add considerable expressive power in many logics, but they do not always increase complexity.