We present a computational screening of experimental structural repositories for fast Li-ion conductors, with the goal of finding new candidate materials for application as solid-state electrolytes in next-generation batteries. We start from ~1400 unique Li-containing materials, of which ~900 are insulators at the level of density-functional theory. For those, we calculate the diffusion coefficient in a highly automated fashion, using extensive molecular dynamics simulations on a potential energy surface (the recently published pinball model) fitted on first-principles forces. The ~130 most promising candidates are studied with full first-principles molecular dynamics, first at high temperature and then more extensively for the 78 most promising candidates. The results of the first-principles simulations of the candidate solid-state electrolytes found are discussed in detail.