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Deep Convolutional Neural Networks (CNNs) are becoming prominent models for semi-automated diagnosis of Alzheimer's Disease (AD) using brain Magnetic Resonance Imaging (MRI). Although being highly accurate, deep CNN models lack transparency and interpretability, precluding adequate clinical reasoning and not complying with most current regulatory demands. One popular choice for explaining deep image models is occluding regions of the image to isolate their influence on the prediction. However, existing methods for occluding patches of brain scans generate images outside the distribution to which the model was trained for, thus leading to unreliable explanations. In this paper, we propose an alternative explanation method that is specifically designed for the brain scan task. Our method, which we refer to as Swap Test, produces heatmaps that depict the areas of the brain that are most indicative of AD, providing interpretability for the model's decisions in a format understandable to clinicians. Experimental results using an axiomatic evaluation show that the proposed method is more suitable for explaining the diagnosis of AD using MRI while the opposite trend was observed when using a typical occlusion test. Therefore, we believe our method may address the inherent black-box nature of deep neural networks that are capable of diagnosing AD.