We determine the number of integral tetrahedra with diameter d, up to isomorphism, for all d ≤ 1000, via computer enumeration. We give an algorithm that enumerates the integral tetrahedra with diameter at most d in O(d⁵) time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral 4×4 matrices with diameter d fulfilling the triangle inequalities we derive an exact formula.