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We propose an interpretable 3D Grid-Attention deep neural network that can accurately predict a person's age and whether they have Alzheimer's disease (AD) from a structural brain MRI scan. Building on a 3D convolutional neural network, we added two attention modules at different layers of abstraction, so that features learned are spatially related to the global features for the task. The attention layers allow the network to focus on brain regions relevant to the task, while masking out irrelevant or noisy regions. In evaluations based on 4,561 3-Tesla T1-weighted MRI scans from 4 phases of the Alzheimer's Disease Neuroimaging Initiative (ADNI), salience maps for age and AD prediction partially overlapped, but lower-level features overlapped more than higher-level features. The brain age prediction network also distinguished AD and healthy control groups better than another state-of-the-art method. The resulting visual analyses can distinguish interpretable feature patterns that are important for predicting clinical diagnosis. Future work is needed to test performance across scanners and populations.