This paper explores the role of trade integration - or openness - for monetary policy transmission in a medium-scale new Keynesian model. Allowing for strategic complementarities in price-setting, we highlight a new dimension of the exchange rate channel by which monetary policy directly impacts domestic inflation: a monetary contraction which appreciates the exchange rate lowers the local currency price of imported goods; this, in turn, induces domestic producers to lower their prices too. We pin down key parameters of the model by matching impulse responses obtained from a vector autoregression on time series for the US relative to the euro area. Our estimation procedure yields plausible parameter values and suggests a strong role for strategic complementarities. Counterfactual simulations show that openness alters monetary transmission significantly. While the contractionary effect of a monetary policy shock on inflation and output tends to increase in openness, we find that monetary policy's control over inflation increases, as the output decline which is necessary to bring about a given reduction of inflation is smaller in more open economies.