We investigate the effect of dust on spiral galaxies by measuring the inclination dependence of optical colours for 24 276 well-resolved Sloan Digital Sky Survey (SDSS) galaxies visually classified via the Galaxy Zoo project. We find clear trends of reddening with inclination which imply a total extinction from face-on to edge-on of 0.7, 0.6, 0.5 and 0.4 mag for the ugri passbands (estimating 0.3 mag of extinction in z band). We split the sample into 'bulgy' (early-type) and 'discy' (late-type) spirals using the SDSS fracdeV (or f(DeV)) parameter and show that the average face-on colour of 'bulgy' spirals is redder than the average edge-on colour of 'discy' spirals. This shows that the observed optical colour of a spiral galaxy is determined almost equally by the spiral type (via the bulge-disc ratio and stellar populations), and reddening due to dust. We find that both luminosity and spiral type affect the total amount of extinction, with discy spirals at M-r similar to -21.5 mag having the most reddening - more than twice as much as both the lowest luminosity and most massive, bulge-dominated spirals. An increase in dust content is well known for more luminous galaxies, but the decrease of the trend for the most luminous has not been observed before and may be related to their lower levels of recent star formation. We compare our results with the latest dust attenuation models of Tuffs et al. We find that the model reproduces the observed trends reasonably well but overpredicts the amount of u-band attenuation in edge-on galaxies. This could be an inadequacy in the Milky Way extinction law (when applied to external galaxies), but more likely indicates the need for a wider range of dust-star geometries. We end by discussing the effects of dust on large galaxy surveys and emphasize that these effects will become important as we push to higher precision measurements of galaxy properties and their clustering.