We have used optical interferometry to obtain multiwavelength visibility curves for eight red giants over the wavelength range 650-1000 nm. The observations consist of wavelength-dispersed fringes recorded with MAPPIT (Masked Aperture-Plane Interference Telescope) at the 3.9-m Anglo-Australian Telescope. We present results for four Miras (R Car, o Cet, R Hya and R Leo) and four semiregular variables (R Dor, W Hya, L2 Pup and γ Cru). All stars except γ Cru show strong variations of angular size with wavelength. A uniform-disc model was found to be a poor fit in most cases, with Gaussian (or other more tapered) profiles preferred. This, together with the fact that most stars showed a systematic increase in apparent size towards the blue and a larger-than-expected linear size, even in the red, all point towards significant scattering by dust in the inner circumstellar environment. Some stars showed evidence for asymmetric brightness profiles, while L2 Pup required a two-component model, indicating an asymmetrical circumstellar dust shell.