Inspired by the need for effective stochastic models to describe the complex behavior of biological motor proteins that move on linear tracks, exact results are derived for the velocity and dispersion of simple linear sequential models (or one-dimensional random walks) with general waiting-time distributions. The concept of "mechanicity" is introduced to conveniently quantify departures from simple "chemical," kinetic rate processes, and its significance is briefly indicated. The results are extended to more elaborate models that have finite side branches and include death processes (to represent the detachment of a motor from the track).