We initiate the study of chromatic numbers for contact graphs of configurations of integer-sized cuboids in three dimensions, all of which are mutually congruent. Disallowing rotations, we show a global upper bound of 8 for the chromatic numbers, which implies that there is a global upper bound of 48 when the cuboids may be rotated freely. Specializing further to cuboids that are required to have a side length of one we obtain more precise upper bounds. Such upper bounds are compared to examples of configurations having relatively large chromatic numbers, leading to a complete determination of some of these chromatic numbers, but in general, the gaps between our upper and lower bounds are rather wide. In particular, we know of no such configuration of any size leading to a chromatic number above 6.