Cine cardiac magnetic resonance imaging (MRI) is widely used for the diagnosis of cardiac diseases thanks to its ability to present cardiovascular features in excellent contrast. As compared to computed tomography (CT), MRI, however, requires a long scan time, which inevitably induces motion artifacts and causes patients' discomfort. Thus, there has been a strong clinical motivation to develop techniques to reduce both the scan time and motion artifacts. Given its successful applications in other medical imaging tasks such as MRI super-resolution and CT metal artifact reduction, deep learning is a promising approach for cardiac MRI motion artifact reduction. In this paper, we propose a novel recurrent generative adversarial network model for cardiac MRI motion artifact reduction. This model utilizes bi-directional convolutional long short-term memory (ConvLSTM) and multi-scale convolutions to improve the performance of the proposed network, in which bi-directional ConvLSTMs handle long-range temporal features while multi-scale convolutions gather both local and global features. We demonstrate a decent generalizability of the proposed method thanks to the novel architecture of our deep network that captures the essential relationship of cardiovascular dynamics. Indeed, our extensive experiments show that our method achieves better image quality for cine cardiac MRI images than existing state-of-the-art methods. In addition, our method can generate reliable missing intermediate frames based on their adjacent frames, improving the temporal resolution of cine cardiac MRI sequences.