Detailed analyses of the microstructure of silver-sheathed (Bi, Pb)2Sr2Ca2Cu3Ox tapes by scanning electron microscopy and X-ray diffraction, as well as measurements of the critical current density and the electrical resistivity of the isolated superconductor cores, have been performed on a series of short samples with filament thicknesses of approx. 40–50 μm and jc (77 K, B=0) ≈ 15 000 A/cm2. The experimental data cannot be satisfactorily explained within the framework of the “brick wall” model for the current transport in highly textured polycrystalline high-temperature superconductors. We therefore propose a new model for Bi/Pb(2223) tapes, based on the frequent small-angle c-axis grain boundaries that serve as highly conducting connections for the current between neighbouring grains, without the necessity for the supercurrent to flow perpendicularly to the CuO planes. The model provides a semi-quantitative description of the field dependence of the critical current density and its anisotropy, and correlates them with characteristics microstructural features of the Bi/Pb(2223) tapes.