All-sky surveys for isolated continuous gravitational waves present a significant data-analysis challenge. Semicoherent search methods are commonly used to efficiently perform the computationally-intensive task of searching for these weak signals in the noisy data of gravitational-wave detectors such as LIGO and Virgo. We present a new implementation of a semicoherent search method, weave, that for the first time makes full use of a parameter-space metric to generate banks of search templates at the correct resolution, combined with optimal lattices to minimize the required number of templates and hence the computational cost of the search. We describe the implementation of weave and associated design choices and characterize its behavior using semianalytic models.