Graph-theoretical methods have rapidly become a standard tool in studies of the structure and function of the human brain. Whereas the structural connectome can be fairly straightforwardly mapped onto a complex network, there are more degrees of freedom in constructing networks that represent functional connections between brain areas. For functional magnetic resonance imaging (fMRI) data, such networks are typically built by aggregating the blood-oxygen-level dependent signal time series of voxels into larger entities (such as Regions of Interest in some brain atlas) and determining their connection strengths from some measure of time-series correlations. Although it is evident that the outcome must be affected by how the voxel-level time series are treated at the preprocessing stage, there is a lack of systematic studies of the effects of preprocessing on network structure. Here, we focus on the effects of spatial smoothing, a standard preprocessing method for fMRI. We apply various levels of spatial smoothing to resting-state fMRI data and measure the changes induced in functional networks. We show that the level of spatial smoothing clearly affects the degrees and other centrality measures of functional network nodes; these changes are non-uniform, systematic, and depend on the geometry of the brain. The composition of the largest connected network component is also affected in a way that artificially increases the similarity of the networks of different subjects. Our conclusion is that wherever possible, spatial smoothing should be avoided when preprocessing fMRI data for network analysis.