Our new IOTA JHK-band beam combiner allows the simultaneous recording of spectrally dispersed J-, H- and K-band Michelson interferograms. In this paper we present our IOTA observations of the Mira star T Cep with this beam combiner (observations in June 2001; four baselines in the range of 14 m to 27 m). The beam combiner optics consists of an anamorphic cylindrical lens system and a prism. From the interferograms of T Cep we derive the visibilities and the J-, H-, and K-band uniform-disk diameters of 14.0 +/- 0.6 mas, 13.7 +/- 0.6 mas and 15.0 +/- 0.6 mas, respectively. Angular stellar filter radii and Rosseland radii are derived from the measured visibilities by fitting theoretical center-to-limb intensity variations (CLVs) of different Mira star models. The available HIPPARCOS parallax (4.76 +/- 0.75 mas) of T Cep allows us to determine linear radii. For example, from the K-band visibility we derive a Rosseland radius of 329-50/+70 solar radii if we use the CLVs of the M-models as fit functions. This radius is in good agreement with the theoretical M-model Rosseland radius of 315 solar radii. The comparison of measured stellar parameters (e.g. diameters, effective temperature, visibility shape) with theoretical parameters indicates whether any of the models is a fair representation of T Cep. The ratios of visibilities of different spectral channels can be measured with higher precision than absolute visibilities. Therefore, we use the visibility ratios V(λ1)/V(λ2) to investigate the wavelength dependence of the stellar diameter. We find that the 2.03 μm uniform-disk diameter of T Cep is about 1.26 times larger than the 2.26 μm uniform-disk diameter.