We study the transmission of monetary policy shocks in a model in which realistic heterogeneity in price rigidity interacts with heterogeneity in sectoral size and input-output linkages, and derive conditions under which these heterogeneities generate large real effects. Quantitatively, heterogeneity in the frequency of price adjustment is the most important driver behind large real effects. Heterogeneity in input-output linkages and consumption shares contribute only marginally to real effects but alter substantially the identity and contribution of the most important sectors to the transmission of monetary shocks. In the model and data, reducing the number of sectors decreases monetary non-neutrality with a similar impact response of inflation. Hence, the initial response of inflation to monetary shocks is not sufficient to discriminate across models and for the real effects of nominal shocks and ignoring heterogeneous consumption shares and input-output linkages identifies the wrong sectors from which the real effects originate.