Compressive Sensing (CS) techniques enable accurate signal reconstruction with few measurements. Deep Unfolding Networks (DUNs) have recently been shown to increase the efficiency of CS by emulating iterative CS optimization procedures by neural networks. However, most of these DUNs suffer from redundant update procedures or complex matrix operations, which can impair their reconstruction performances. Here we propose the optimization-inspired Cumulative Transmission Network (CT-Net), a DUN approach for natural image CS. We formulate an optimization procedure introducing an auxiliary variable similar to Half Quadratic Splitting (HQS). Unfolding this procedure defines the basic structure of our neural architecture, which is then further refined. A CT-Net is composed of Reconstruction Fidelity Modules (RFMs) for minimizing the reconstruction error and Constraint Gradient Approximation (CGA) modules for approximating (the gradient of) sparsity constraints instead of relying on an analytic solutions such as soft-thresholding. Furthermore, a lightweight Cumulative Transmission (CT) between CGAs in each reconstruction stage is proposed to facilitate a better feature representation. Experiments on several widely used natural image benchmarks illustrate the effectiveness of CT-Net with significant performance improvements and fewer network parameters compared to existing state-of-the-art methods. The experiments also demonstrate the scene and noise robustness of the proposed method.