The Ccr4-Not complex removes mRNA poly(A) tails to regulate eukaryotic mRNA stability and translation. RNA-binding proteins contribute to specificity by interacting with both Ccr4-Not and target mRNAs, but this is not fully understood. Here, we reconstitute accelerated and selective deadenylation of RNAs containing AU-rich elements (AREs) and Pumilio-response elements (PREs). We find that the fission yeast homologues of Tristetraprolin/TTP and Pumilio/Puf (Zfs1 and Puf3) interact with Ccr4-Not via multiple regions within low-complexity sequences, suggestive of a multipartite interface that extends beyond previously defined interactions. Using a two-color assay to simultaneously monitor poly(A) tail removal from different RNAs, we demonstrate that Puf3 can distinguish between RNAs of very similar sequence. Analysis of binding kinetics reveals that this is primarily due to differences in dissociation rate constants. Consequently, motif quality is a major determinant of mRNA stability for Puf3 targets in vivo and can be used for the prediction of mRNA targets.