Periodically poled thin-film lithium niobate (TFLN) waveguides have emerged as a leading platform for highly efficient frequency conversion in the near-infrared. However, the commonly used silica bottom-cladding results in high absorption loss at wavelengths beyond 2.5 µm. In this work, we demonstrate efficient frequency conversion in a TFLN-on-sapphire platform, which features high transparency up to 4.5 µm. In particular, we report generating mid-infrared light up to 3.66 µm via difference-frequency generation of a fixed 1 µm source and a tunable telecom source, with normalized efficiencies up to 200%/Wcm^2. These results show TFLN-on-sapphire to be a promising platform for integrated nonlinear nanophotonics in the mid-infrared.