Alzheimer's disease affects over 55 million people worldwide and is projected to more than double by 2050, necessitating rapid, accurate, and scalable diagnostics. However, existing approaches are limited because they cannot achieve clinically acceptable accuracy, generalization across datasets, robustness to missing modalities, and explainability all at the same time. This inability to satisfy all these requirements simultaneously undermines their reliability in clinical settings. We propose OmniBrain, a multimodal framework that integrates brain MRI, radiomics, gene expression, and clinical data using a unified model with cross-attention and modality dropout. OmniBrain achieves $92.2 \pm 2.4\%$accuracy on the ANMerge dataset and generalizes to the MRI-only ADNI dataset with $70.4 \pm 2.7\%$ accuracy, outperforming unimodal and prior multimodal approaches. Explainability analyses highlight neuropathologically relevant brain regions and genes, enhancing clinical trust. OmniBrain offers a robust, interpretable, and practical solution for real-world Alzheimer's diagnosis.