Recent advances in Dynamic Sparse Training (DST) have pushed the frontier of sparse neural network training in structured and unstructured contexts, matching dense-model performance while drastically reducing parameter counts to facilitate model scaling. However, unstructured sparsity often fails to translate into practical speedups on modern hardware. To address this shortcoming, we propose DynaDiag, a novel structured sparse-to-sparse DST method that performs at par with unstructured sparsity. DynaDiag enforces a diagonal sparsity pattern throughout training and preserves sparse computation in forward and backward passes. We further leverage the diagonal structure to accelerate computation via a custom CUDA kernel, rendering the method hardware-friendly. Empirical evaluations on diverse neural architectures demonstrate that our method maintains accuracy on par with unstructured counterparts while benefiting from tangible computational gains. Notably, with 90\% sparse linear layers in ViTs, we observe up to a 3.13x speedup in online inference without sacrificing model performance and a 1.59x speedup in training on a GPU compared to equivalent unstructured layers.