The dielectric and non-Ohmic properties of Na1/2Y1/2Cu3Ti4O12 ceramics sintered under various conditions to obtain different microstructures were investigated. Microstructure analysis confirmed the presence of Na, Y, Cu, Ti, and O and these elements were well dispersed in the microstructure. Na1/2Y1/2Cu3Ti4O12 ceramics exhibited non-Ohmic characteristics with large nonlinear coefficients of about 5.7-6.6 irrespectively of sintering conditions. The breakdown electric field of fine-grained ceramic with the mean grain size of approximate to 1.7 m (approximate to 5600 V/cm) was much larger than those of the course-grained ceramics with grain sizes of approximate to 9.5-10.4 m (approximate to 1850-2180 V/cm). Through optimization of sintering conditions, a low loss tangent of about 0.03 and very high dielectric permittivities of 18 000-23 000 with good temperature stability were successfully accomplished. The electrical responses of the grains and grain boundaries can, respectively, be well described using admittance and impedance spectroscopy analyses based on the brickwork layer model. A possible mechanism for the origin of semiconducting grains is discussed. The colossal dielectric response was reasonably described as closely correlated with the electrically heterogeneous microstructure by means of strong interfacial polarization at the insulating grain-boundary layers. The non-Ohmic properties of Na1/2Y1/2Cu3Ti4O12 ceramics were primarily related to their microstructure, i.e., grain size and volume fraction of grain boundaries.