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. Empirically, heterogeneity in the frequency of price adjustment is the most important driver behind large real effects, whereas heterogeneity in input-output linkages contributes only marginally, with differences in consumption shares in between. Heterogeneity in price rigidity further is key in determining which sectors are the most important contributors to the transmission of monetary shocks, and is necessary but not sufficient to generate realistic output correlations. 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.