Background: Although convolutional neural networks (CNN) achieve high diagnostic accuracy for detecting Alzheimers disease (AD) dementia based on magnetic resonance imaging (MRI) scans, they are not yet applied in clinical routine. One important reason for this is a lack of model comprehensibility. Recently developed visualization methods for deriving CNN relevance maps may help to fill this gap. We investigated whether models with higher accuracy also rely more on discriminative brain regions predefined by prior knowledge. Methods: We trained a CNN for the detection of AD in N=663 T1-weighted MRI scans of patients with dementia and amnestic mild cognitive impairment (MCI) and verified the accuracy of the models via cross-validation and in three independent samples including N=1655 cases. We evaluated the association of relevance scores and hippocampus volume to validate the clinical utility of this approach. To improve model comprehensibility, we implemented an interactive visualization of 3D CNN relevance maps. Results: Across three independent datasets, group separation showed high accuracy for AD dementia vs. controls (AUC$geq$0.92) and moderate accuracy for MCI vs. controls (AUC$approx$0.75). Relevance maps indicated that hippocampal atrophy was considered as the most informative factor for AD detection, with additional contributions from atrophy in other cortical and subcortical regions. Relevance scores within the hippocampus were highly correlated with hippocampal volumes (Pearsons r$approx$-0.86, p<0.001). Conclusion: The relevance maps highlighted atrophy in regions that we had hypothesized a priori. This strengthens the comprehensibility of the CNN models, which were trained in a purely data-driven manner based on the scans and diagnosis labels.