The transport properties of block copolymer-derived polymer electrolyte membranes (PEMs) are sensitive to microstructural disorder originating in the randomly oriented microdomains produced during uncontrolled self-assembly by microphase separation. This microstructural disorder can negatively impact performance due to the presence of conductivity-impeding grain boundaries and the resulting tortuosity of transport pathways. We use magnetic fields to control the orientational order of Li-doped lamellar polyethylene oxide (PEO) microdomains in a liquid crystalline diblock copolymer over large length scales (>3 mm). Microdomain alignment results in an increase in the conductivity of the membrane, but the improvement relative to non-aligned samples is modest, and limited to roughly 50% in the best cases. This limited increase is in stark contrast to the order of magnitude improvement observed for magnetically aligned cylindrical microdomains of PEO. Further, the temperature dependence of the conductivity of lamellar microdomains is seemingly insensitive to the order-disorder phase transition, again in marked contrast to the behavior of cylinder-forming materials. The data are confronted with theoretical predictions of the microstructural model developed by Sax and Ottino. The disparity between the conductivity enhancements obtained by domain alignment of cylindrical and lamellar systems is rationalized in terms of the comparative ease of percolation due to the intersection of randomly oriented lamellar domains (2D sheets) versus the quasi-1D cylindrical domains. These results have important implications for the development of methods to maximize PEM conductivity in electrochemical devices, including batteries.