Predictive processing is emerging as a common neurocomputational hypothesis to account for diverse psychological functions subserved by a brain, enabling characterization of computational capacities of its distinct substructures. However, the literature is currently lacking a systems-level framework for understanding brain structure-function relationships based on the unified computational principles. Here, we contribute to this framework by examining gradients of functional connectivity as a low dimensional spatial representation of functional variation within a given structure. Specifically, we investigated functional connectivity gradients in the cerebral cortex, the cerebellum, and the hippocampus using resting-state functional MRI data collected from large samples of healthy young adults. We then evaluated the degree to which these structures share common principles of functional organization by assessing the correspondence of their gradients. We show that the organizing principles of these structures primarily follow two functional gradients consistent with the existing accounts of predictive processing: A model-error gradient that describes the flow of prediction and prediction error signals, and a model-precision gradient that differentiates regions involved in the representation and attentional modulation of such signals in the cerebral cortex. Using these gradients, we also demonstrate triangulation of functional connectivity involving distinct subregions of the three structures, suggesting the existence of parallel functional circuits that may subserve different aspects of predictive processing in the brain. These findings allow formulation of novel computational hypotheses about the functional relationships between the cerebral cortex, the cerebellum, and the hippocampus that may be instrumental for understanding the brain's dynamics within its large-scale predictive architecture.