Abstract: The unipotent subgroup of a finite group of Lie type over a prime field Z/pZ comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present paper, we show that the diameter of this Cayley graph is bounded above and below by constant multiples of np + n^2 log p, where n is the rank of the associated Lie group. This generalizes a result of the first author, which treated the case of SL_n(Z/pZ). (Keywords: diameter, Cayley graph, finite groups of Lie type. AMS classification: 20G40, 05C25)