Tracking multiple moving targets is known to be challenged by the nonlinearity present in the measurement equation, and by the computationally burdensome data association task. This paper introduces a grid-based model of target signal strengths leading to linear state and measurement equations, that can afford state estimation via sparsity-aware Kalman filtering (KF), and bypasses data association. Leveraging the sparsity inherent to the novel grid-based model, a sparsity-cognizant KF tracker is developed that effects sparsity through ℓ1-norm regularization. The proposed tracker does not require knowledge of the number of targets or their signal strengths, and exhibits considerably lower complexity than the hidden Markov filter benchmark, especially as the number of targets increases. Numerical simulations demonstrate that the sparsity-cognizant tracker enjoys improved root mean-square error performance at reduced complexity when compared to its sparsity-agnostic counterparts.