Kronecker compressed sensing refers to using Kronecker product matrices as sparsifying bases and measurement matrices in compressed sensing. This work focuses on the Kronecker compressed sensing problem, encompassing three sparsity structures: $(i)$ a standard sparsity model with arbitrarily positioned nonzero entries, $(ii)$ a hierarchical sparsity model where nonzero entries are concentrated in a few blocks, each with only a subset of nonzero entries, and $(iii)$ a Kronecker-supported sparsity model where the support vector is a Kronecker product of smaller vectors. We present a hierarchal view of Kronecker compressed sensing that explicitly reveals a multiple-level sparsity pattern. This framework allows us to utilize the Kronecker structure of dictionaries and design a two-stage sparse recovery algorithm for different sparsity models. Further, we analyze the restricted isometry property of Kronecker-structured matrices under different sparsity models. Simulations show that our algorithm offers comparable recovery performance to state-of-the-art methods while significantly reducing runtime.