We use a data set of nearby galaxies drawn from the HERA CO-Line Extragalactic Survey, ATLAS(3D), and COLD GASS surveys to study variations in molecular gas depletion time (t(dep)) in galaxy structures such as bulges, grand-design spiral arms, bars and rings. Molecular gas is traced by CO line emission and star formation rate (SFR) is derived using the combination of far-ultraviolet and mid-infrared (MIR) data. The contribution of old stars to MIR emission for the ATLAS(3D) sample is corrected using 2MASS K-band images. We apply a two-dimensional image decomposition algorithm to decompose galaxies into bulges and discs. Spiral arms, bars and rings are identified in the residual maps, and molecular gas depletion times are derived on a square grid of 1 kpc(2) size. In previous work, we showed that t(dep) correlates strongly with specific star formation rate (sSFR). We now find that at a given sSFR, the bulge has shorter t(dep) than the disc. The shift to shorter depletion times is most pronounced in the inner bulge (R < 0.1 R-e). Grids from galaxies with bars and rings are similar to those from galactic bulges in that they have reduced t(dep) at a given sSFR. In contrast, the t(dep) versus sSFR relation in the discs of galaxies with spiral arms is displaced to longer t(dep) at fixed sSFR. We then show that the differences in the t(dep)-sSFR relation for bulges, discs, arms, bars and rings can be linked to variations in stellar, rather than gas surface density between different structures. Our best current predictor for t(dep), both globally and for 1 kpc grids, is given by t(dep) = -0.36 log (Sigma(SFR)) -0.50 log (Sigma(*)) + 5.87.