Of the five regular platonic solids, only the cube tiles Euclidean space. The authors consider tile-transitive tilings by topologically platonic solids, i.e. by topological polyhedra that are homeomorphic to one of the platonic solids, and such that the group of the tiling is a crystallographic space group. The authors show that there are 914, 58, and 46 equivariant types of tile-transitive tilings of Euclidean space by topological cubes, octahedra and tetrahedra, and that these fall into 11, 3, and 9 topological families, respectively. The classification is obtained with the help of a combinatorial algorithm based on Delaney symbols. Each topological family is illustrated by a picture accompanied by a diagram that gives a precise description of the tiling.