RNA degradation is a tightly regulated and highly conserved process which selectively targets aberrant RNAs using both 5’ and 3’ exonucleases. The RNAs degraded by this process include viral RNA, but the mechanisms by which viral RNA is identified and recruited to the degradation machinery are incompletely understood. To identify new antiviral genes, we performed RNAi screening of genes with known roles in RNA metabolism in Drosophila cells. We identified the RNA exosome, which targets RNA for 3’ end decay, and two components of the exosome cofactor TRAMP complex, dMtr4 and dZcchc7, as antiviral against a panel of RNA viruses. As these genes are highly conserved, I extended these studies to human cells and found that the exosome as well as TRAMP components hMTR4 and hZCCHC7 are antiviral. While hMTR4 and hZCCHC7 are normally nuclear, I found that infection by cytoplasmic RNA viruses induces their export to cytoplasmic granules, where they form a complex that specifically recognizes and induces degradation of viral mRNAs. Furthermore, I found that the 3’ UTR of bunyaviral mRNA is sufficient to confer virus-induced exosomal degradation, demonstrating cis-regulation. Several types of ribonucleoprotein (RNP) granules interact with both 5’ and 3’ decay machinery to facilitate degradation of sequestered RNAs. In order to determine whether TRAMP component-containing granules contain components of other defined RNP granules, I performed immunofluorescence for hZCCHC7 as well as components of P-bodies, stress granules, and exosome granules and found that hZCCHC7 can colocalize with proteins resident in exosome granules and stress granules during viral infection, suggesting that hZCCHC7 may bind translationally-stalled viral RNAs and bring them to exosome granules for degradation. To further characterize the regulation of TRAMP component nuclear export during infection, I investigated the viral signals necessary for this transport. I found that transfection with dsRNA is sufficient to induce relocalization, while infection with UV-inactivated viruses is not. Moreover, I tested the role of canonical innate immune adaptors in this process and found that the dsRNA sensor PKR promoted relocalization during Sindbis virus infection. Altogether, my results reveal that the presence of replicating viral RNA causes TRAMP components to be repurposed to a cytoplasmic surveillance role in several classes of RNP granules including stress granules and exosome granules. There, they selectively engage viral RNAs for degradation to restrict a broad range of viruses.