The strong spin-spin exchange interaction in some low-dimensional magnetic materials can give rise to a high group velocity and thermal conductivity contribution from magnons. One example is the incommensurate layered compounds (Sr,Ca,La)14Cu24O41. The effects of grain boundaries and defects on quasi-one-dimensional magnon transport in these compounds are not well understood. Here we report the microstructures and anisotropic thermal transport properties of textured Sr14Cu24O41, which are prepared by solid-state reaction followed by spark plasma sintering. Transmission electron microscopy clearly reveals nano-layered grains and the presence of dislocations and planar defects. The thermal conductivity contribution and mean free paths of magnons in the textured samples are evaluated with the use of a kinetic model for one-dimensional magnon transport, and found to be suppressed significantly as compared to single crystals at low temperatures. The experimental results can be explained by a one-dimensional magnon-defect scattering model, provided that the magnon-grain boundary scattering mean free path in the anisotropic magnetic structure is smaller than the average length of these nano-layers along the c axis. The finding suggests low transmission coefficients for magnons across grain boundaries.