Amyloid-$\beta$ (A$\beta$) plaques in conjunction with hyperphosphorylated tau proteins in the form of neurofibrillary tangles are the two neuropathological hallmarks of Alzheimer's disease (AD). In particular, the accumulation of A$\beta$ plaques, as evinced by the A/T/N (amyloid/tau/neurodegeneration) framework, marks the initial stage. Thus, the identification of individuals with A$\beta$ positivity could enable early diagnosis and potentially lead to more effective interventions. Deep learning methods relying mainly on amyloid PET images have been employed to this end. However, PET imaging has some disadvantages, including the need of radiotracers and expensive acquisitions. Hence, in this work, we propose a novel multimodal approach that integrates information from structural, functional, and diffusion MRI data to discriminate A$\beta$ status in the AD continuum. Our method achieved an accuracy of $0.762\pm0.04$. Furthermore, a \textit{post-hoc} explainability analysis (guided backpropagation) was performed to retrieve the brain regions that most influenced the model predictions. This analysis identified some key regions that were common across modalities, some of which were well-established AD-discriminative biomarkers and related to A$\beta$ deposition, such as the hippocampus, thalamus, precuneus, and cingulate gyrus. Hence, our study demonstrates the potential viability of MRI-based characterization of A$\beta$ status, paving the way for further research in this domain.