Consider the system of delay difference equations x(i)(t) - x(i)(t - sigma) +(k = 1)Sigma(l) (j = 1)Sigma(n) p(ijk)x(j)(t -tau(k)) = 0, (*) where p(ijk) is an element of (-infinity,infinity), sigma and tau(k) is an element of (0,infinity), i,j = 1,2,...,n,k = 1,2,...,l. Sufficient conditions are obtained for all solutions of the system (*) to be oscillatory. (C) 1999 Academic Press.