An important component of the mammalian stress response is the reprogramming of translation. A variety of stresses trigger abrupt polysome disassembly and the accumulation of stalled translation preinitiation complexes. These complexes nucleate cytoplasmic stress granules (SGs), sites of mRNA triage in which mRNAs from disassembling polysomes are sorted and the fates of individual transcripts are determined. Here, we demonstrate that influenza A virus (IAV) actively suppresses SG formation during infection, thereby allowing translation of viral mRNAs. Complete inhibition of SG formation is dependent on the function of the viral nonstructural protein 1 (NS1); at late times postinfection, cells infected with NS1-mutant viruses formed SGs in a double-stranded RNA-activated protein kinase (PKR)-dependent fashion. In these cells, SG formation correlated with inhibited viral protein synthesis. Together, these experiments demonstrate the antiviral potential of SGs and reveal a viral countermeasure that limits SG formation.