We propose novel methods to identify tweets that criticize partisan news sources. Prior work suggests that criticism, ridicule, and distrust of news media all play important roles in hyperpartisanship, misinformation, and filter bubble formation. Thus, understanding the prevalence and temporal dynamics of media-targeted criticism can provide us with updated tools to assess the health of the information ecosystem. There is a scarcity of labeled data for this task, and we develop a weakly supervised learning approach that leverages multiple noisy labeling functions based on both the content of the tweet as well as the historical news sharing behavior of the user. Using this classifier, we explore how tweets expressing criticism vary by user, news source, and time, finding substantial spikes in media criticism during politically polarizing events, such as the investigation into Russian interference in the 2016 U.S.~elections and the 2017 ``unite the right'' rally in Charlottesville. This type of media-targeting criticism is also more likely to occur after users have been exposed to unreliable and hyperpartisan media.