While ultrasound imaging has seen vast technical advances over the last decades, trans-thoracic echocardiography still suffers from image quality degradation caused by acoustic interaction with inhomogeneous tissue layers between the transducer and the heart. The acoustic energy reflections from echogenic structures such as skin, subcutaneous fat, bone, cartilage, intercostal muscle tissue, and lungs can form a dense overlay of echoes occluding the structural information resulting in a degradation of the diagnostic value. We propose a new method for reducing this reverberational clutter inspired by how the brain addresses the problem; identifying the reverberation overlay by the way it constitutes a pattern of speckles that moves in one cohesive motion different from that of the underlying structures. With this approach, we effectively render the clutter suppression as a video separation problem. Compared to traditional clutter rejection methods that tend to specialize in either temporal or spatial qualities, we find a neural network to be more flexible in incorporating both temporal and spatial information. We generate a pseudo-paired data set using in vivo data by excising patches off hypo-echoic regions of strongly reverberation-affected clinical recordings and superimposing them onto clean clinical recordings. The pseudo-paired data set of beamformed in-phase and quadrature component (IQ)-data is used to train a neural network to suppress reverberations in cine-loops. We demonstrate that this post-beamformer method can enhance image quality in in vivo and make valuable clinical structures clearer in a commercial system. We show that the method does not display any tendency to generate false cardiac structures, and that rapid motions from e.g. valve leaflets retain high structural integrity and low levels of blurring. Our results suggest that this method can be an effective and robust tool for suppressing reverberations in transthoracic ultrasound imaging.