An explicit multirate piecewise constant sampled state feedback (MPCSSF) control for linear periodic continuous-time systems is presented, which is based on a revised generalized reachability gramian first introduced by Al-Rahmani and Franklin. Using MPCSSF control, the LQR optimal problem for linear periodic systems is reduced to the LQR optimal problem for discrete-time linear periodic systems. It is shown that there is a unique optimal solution, which calls for solving a difference periodic Riccati equation, iff the assumptions of controllability (stabilizability) and observability (detectability) on the original system hold. The MPCSSF control can be viewed as an extension of Al-Rahmani and Franklin's method to the case of multirate state sampling.