The pulmonary capillary networks (PCNs) embody organ-specific microvasculatures, where blood vessels form dense meshes that maximize the surface area available for gas exchange in the lungs. With characteristic capillary lengths and diameters similar to the size of red blood cells (RBCs), seminal descriptions coined the term “sheet flow” nearly half a century ago to differentiate PCNs from the usual notion of Poiseuille flow in long straight tubes. Here, we revisit in true-scale experiments the original “sheet flow” model and devise for the first time biomimetic microfluidic platforms of organ-specific PCN structures perfused with RBC suspensions at near-physiological hematocrit levels. By implementing RBC tracking velocimetry, our measurements reveal a wide range of heterogonous RBC pathways that coexist synchronously within the PCN; a phenomenon that persists across the broad range of pressure drops and capillary segment sizes investigated. Interestingly, in spite of the intrinsic complexity of the PCN structure and the heterogeneity in RBC dynamics observed at the microscale, the macroscale bulk flow rate versus pressure drop relationship retains its linearity, where the hydrodynamic resistance of the PCN is to a first order captured by the characteristic capillary segment size. To the best of our knowledge, our in vitro efforts constitute a first, yet significant, step in exploring systematically the transport dynamics of blood in morphologically inspired capillary networks.