In response to viral infection, mammalian cells activate several innate immune pathways to antagonize viral gene expression. Upon recognition of viral double-stranded RNA, protein kinase R (PKR) phosphorylates the alpha subunit of eukaryotic initiation factor 2 (eIF2α) on serine 51. This inhibits canonical translation initiation, which broadly antagonizes viral protein synthesis. It also promotes the assembly of cytoplasmic ribonucleoprotein complexes termed stress granules (SGs). SGs are widely thought to promote cell survival and antiviral signaling. However, co-activation of the OAS/RNase L antiviral pathway inhibits the assembly of SGs and promotes the assembly of an alternative ribonucleoprotein complex termed an RNase L-dependent body (RLB). The formation of RLBs has been observed in response to double-stranded RNA, dengue virus infection, or SARS-CoV-2 infection. Herein, we review the distinct biogenesis pathways and properties of SGs and RLBs, and we provide perspective on their potential functions during the antiviral response.