The convergence of organoid technology and three-dimensional (3D) bioprinting has improved control over human tissue models, yet spatial precision at fabrication does not recapitulate the dynamic processes by which tissues develop, remodel, respond to injury, and recover. Here, we frame multidimensional bioprinting by capability rather than as a mature 3D, 4D, 5D, 6D hierarchy. The framework comprises spatial patterning, temporal programming, curvature-aware deposition, environmental responsiveness, and feedback. Direct organoid evidence is strongest for spatially controlled 3D printing and selected temporally programmed systems; multi-axis curvature control is supported mainly by adjacent biofabrication studies, whereas 6D remains a non-standardized and largely conceptual integration of multi-axis fabrication with stimulus responsiveness. We distinguish direct organoid-bioprinting evidence from enabling technologies and conceptual proposals, and evaluate each added capability against matched static 3D controls, longitudinal measurements, and organ-specific functional outcomes. This framework links engineering control to morphogenesis, disease trajectories, and adaptive organoid platforms, while emphasizing that progress should be judged by mechanistic interpretability, predictive performance, and interventional control rather than by dimensional labels alone.