We consider the problem of finding the optimal time-periodic sensor schedule for field estimation in wireless sensor networks. The sensors are subject to resource constraints in the form of a limit on the number of times each sensor can be activated over one period. A correspondence between active sensors and the nonzero columns of the estimator gain matrix is established. We formulate an optimization problem which seeks the optimal periodic estimator gain matrix that minimizes the error covariance over one period, subject to the sensor resource constraints. The optimization problem is combinatorial in nature, and we apply the alternating direction method of multipliers (ADMM) to find its locally optimal solutions. Numerical results are provided to illustrate the effectiveness of the proposed method.