The present invention is directed to membranes composed liquid crystals having continuous aqueous channels, such as a lyotropic liquid crystal, including a cubic phase lyotropic liquid crystal, and to electrochemical sensors equipped with such membranes. The membranes are useful in limiting the diffusion of an analyte to a working electrode in an electrochemical sensor so that the sensor does not saturate and/or remains linearly responsive over a large range of analyte concentrations. Electrochemical sensors equipped with membranes of the present invention demonstrate considerable sensitivity and stability, and a large signal-to-noise ratio, in a variety of conditions.