This paper studies the classic problem of finding heavy hitters in the turnstile streaming model. We give the first deterministic linear sketch that has $O(ε^{-2} \log n \cdot \log^*(ε^{-1}))$ rows and answers queries in sublinear time. The number of rows is only a factor of $\log^*(ε^{-1})$ more than that used by the state-of-the-art algorithm prior to our paper due to Nelson, Nguyen and Woodruff (RANDOM'12). Their algorithm runs in time at least linear in the universe size $n$, which is highly undesirable in streaming applications. Our approach is based on an iterative procedure, where most unrecovered heavy hitters are identified in each iteration. Although this technique has been extensively employed in the related problem of sparse recovery, this is the first time, to the best of our knowledge, that it has been used in the context of $\ell_1$ heavy hitters. Along the way, we also give sublinear time algorithms for the closely related problems of combinatorial group testing and $\ell_1/\ell_1$ compressed sensing, matching the space usage of previous (super-)linear time algorithms.