Obtaining accurate and reliable images from low-dose computed tomography (CT) is challenging. Regression convolutional neural network (CNN) models that are learned from training data are increasingly gaining attention in low-dose CT reconstruction. This paper modifies the architecture of an iterative regression CNN, BCD-Net, for fast, stable, and accurate low-dose CT reconstruction, and presents the convergence property of the modified BCD-Net. Numerical results with phantom data show that applying faster numerical solvers to model-based image reconstruction (MBIR) modules of BCD-Net leads to faster and more accurate BCD-Net; BCD-Net significantly improves the reconstruction accuracy, compared to the state-of-the-art MBIR method using learned transforms; BCD-Net achieves better image quality, compared to a state-of-the-art iterative NN architecture, ADMM-Net. Numerical results with clinical data show that BCD-Net generalizes significantly better than a state-of-the-art deep (non-iterative) regression NN, FBPConvNet, that lacks MBIR modules.