Although microbial communities are ubiquitous in nature, relatively little is known about the structural and functional roles of their constituent organisms' underlying interactions. A common approach to study such questions begins with extracting a network of statistically significant pairwise co-occurrences from a matrix of observed operational taxonomic unit (OTU) abundances across sites. The structure of this network is assumed to encode information about ecological interactions and processes, resistance to perturbation, and the identity of keystone species. However, common methods for identifying these pairwise interactions can contaminate the network with spurious patterns that obscure true ecological signals. Here, we describe this problem in detail and develop a solution that incorporates null models to distinguish ecological signals from statistical noise. We apply these methods to the initial OTU abundance matrix and to the extracted network. We demonstrate this approach by applying it to a large soil microbiome data set and show that many previously reported patterns for these data are statistical artifacts. In contrast, we find the frequency of three-way interactions among microbial OTUs to be highly statistically significant. These results demonstrate the importance of using appropriate null models when studying observational microbiome data, and suggest that extracting and characterizing three-way interactions among OTUs is a promising direction for unraveling the structure and function of microbial ecosystems.