We report results of the flow of dilute mono-disperse solutions of atactic poly(styrene) in di-octyl phthalate through regular crystallographic packed beds of spheres. Pressure drop measurements made as a function of flow rate across simple cubic and body centred cubic arrays of spheres have been used to estimate the specific viscosities of the polymer solutions as a function of the superficial Deborah number. Through both structures the onset Deborah number for the non-Newtonian increase in specific viscosity is found to be low when compared on the basis of well-characterized Zimm relaxation times. Surprisingly it is found that polymer solutions achieve a greater maximum specific viscosity in the simple cubic than in the body centred cubic array, a result contrary to prior expectations due to the absence of trailing stagnation points in the simple cubic structure. It is hypothesised that the trailing stagnation points in the body centred cubic array may be screened from the flow field by strands of oriented polymer and that, as such, the periodic variations in cross-sectional area of the flow (which are more severe in the simple cubic array) may play the most significant role in causing polymer extension and in enhancing the non-Newtonian viscosity.