The ability to monitor and perturb RNAs in living cells would benefit greatly from a modular protein architecture that targets unmodified RNA sequences in a programmable way. We report that the RNA-binding protein PumHD (Pumilio homology domain), which has been widely used in native and modified form for targeting RNA, can be engineered to yield a set of four canonical protein modules, each of which targets one RNA base. These modules (which we call Pumby, for Pumilio-based assembly) can be concatenated in chains of varying composition and length, to bind desired target RNAs. The specificity of such Pumby–RNA interactions was high, with undetectable binding of a Pumby chain to RNA sequences that bear three or more mismatches from the target sequence. We validate that the Pumby architecture can perform RNA-directed protein assembly and enhancement of translation of RNAs. We further demonstrate a new use of such RNA-binding proteins, measurement of RNA translation in living cells. Pumby may prove useful for many applications in the measurement, manipulation, and biotechnological utilization of unmodified RNAs in intact cells and systems.