Simple explicit solutions of the linear wave equation in three dimensions are presented which describe wave packets exponentially localized near a point moving with the wave speed. For large values of a certain free parameter these new solutions are localized in Gaussian manner with respect to longitudinal and transverse variables and time. This agrees with considerations by Babich-Ulin and Ralston who have presented an asymptotic description of solutions exhibiting such local behavior. Global estimates and large-time asymptotics of these solutions are given.