In this paper, we present a Lattice Boltzmann scheme for diffusion on unstructured triangular grids. In this formulation there is no need for interpolation, as is required in other LB schemes on irregular grids. At the end of the propagation step, the lattice gas particles arrive exactly at neighbouring lattice sites, as is the case in LB schemes on Bravais lattices. The scheme is constructed using the constraints that the moments of the equilibrium distribution equals that of the Maxwell-Boltzmann distribution. For a special choice of the relaxation parameter (omega = 1) we show that our LB scheme is identical to a cell-centred Finite Volume scheme on an unstructured triangular grid. Consequences of the use of the unstructured grid on the required data structures is discussed in detail. Subsequent simulation show that diffusion on this unstructured grid is isotropic. (C) 2003 Elsevier B.V. All rights reserved.