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Linear recurrent networks (LRNNs) and linear state space models (SSMs) promise computational and memory efficiency on long-sequence modeling tasks, yet their diagonal state transitions limit expressivity. Dense and nonlinear architectures (e.g., LSTMs) on the other hand are provably more expressive, but computationally costly. Here, we explore how expressivity in LRNNs can be increased via richer state mixing across time and channels while maintaining competitive efficiency. Specifically, we introduce two structured LRNN architectures: (i) Higher-order Linear Recurrent Units (H-LRU), which generalize first-order recurrence to higher order, mixing multiple past states, and (ii) Block-Diagonal LRUs (BD-LRU), which enable dense intra-block channel mixing. Per-channel (H-LRU) or per-row (BD-LRU) L1-normalization of selective gates stabilizes training and allows for scaling window/block sizes. A parallel-scan implementation of the proposed architectures keeps the throughput competitive with diagonal LRNNs for moderate orders (H-LRU) and block sizes (BD-LRU). In synthetic sequence modeling tasks, the performance of BD-LRU matches or exceeds those of linear SSMs (Mamba), low-rank LRNNs (DeltaNet) and LSTM baselines, while H-LRU is found to be the most parameter-efficient in compression task. In both synthetic sequence modeling and language modeling, our results indicate that the structure of state mixing rather than width alone shapes expressivity of LRNNs, offering a practical route to closing the efficiency-expressivity gap in linear sequence models.