Spectra are presented of 21 planetary nebulae spanning the well-known, but still unidentified, narrow emission features at 3.2-3.6 mum. The equivalent width of the 3.29-μm emission band is strongly correlated with the gas-phase carbon-to-oxygen ratio, as expected for a band origin in carbon-rich grains or molecules. It displays an approximately linear dependence upon the C/O ratio, with a threshold near C/O ~0.6. The emission band is present in 18 of the 21 nebulae, being absent in three of the six oxygen-rich objects and only weakly present in a fourth. The profile of the 3.29-μm band is closely similar in all of the nebulae and distinct from the band profile seen in some stellar envelopes. Weaker emission features in the 3.4-3.5 μm region are detected in nine objects, and are prominent only in nitrogen-rich type I nebulae. The intensity of the 3.4-μm feature in these objects is correlated with that of the 3.29-μm band, while the ratio of the two bands is strongly correlated with the nitrogen abundance. Possible explanations for this behaviour are discussed.