Certain planetary nebulae (PNe) contain shells, filaments, or globules of cold gas and dust whose heating and chemistry are likely driven by UV and X-ray emission from their central stars and from wind-collision-generated shocks. We present the results of a survey of molecular line emission in the 88–236 GHz range from nine nearby (<1.5 kpc) planetary nebulae spanning a range of UV and X-ray luminosities, using the 30 m telescope of the Institut de Radioastronomie Millimétrique. Rotational transitions of thirteen molecules, including CO isotopologues and chemically important trace species, were observed and the results compared with and augmented by previous studies of molecular gas in PNe. Lines of the molecules HCO+, HNC, HCN, and CN, which were detected in most objects, represent new detections for four planetary nebulae in our study. Specifically, we present the first detections of 13CO (1–0, 2–1), HCO+, CN, HCN, and HNC in NGC 6445; HCO+ in BD+30°3639; 13CO (2–1), CN, HCN, and HNC in NGC 6853; and 13CO (2–1) and CN in NGC 6772. Flux ratios were analyzed to identify correlations between the central star and/or nebular UV and X-ray luminosities and the molecular chemistries of the nebulae. This analysis reveals a surprisingly robust dependence of the HNC/HCN line ratio on PN central star UV luminosity. There exists no such clear correlation between PN X-rays and various diagnostics of PN molecular chemistry. The correlation between HNC/HCN ratio and central star UV luminosity demonstrates the potential of molecular emission line studies of PNe for improving our understanding of the role that high-energy radiation plays in the heating and chemistry of photodissociation regions.