We consider the Tutte polynomial of three classes of greedoids: those arising from rooted graphs, rooted digraphs and binary matrices. We establish the computational complexity of evaluating each of these polynomials at each fixed rational point (x,y). In each case we show that evaluation is #P-hard except for a small number of exceptional cases when there is a polynomial time algorithm. In the binary case, establishing #P-hardness along one line relies on Vertigan's unpublished result on the complexity of counting bases of a matroid. For completeness, we include an appendix providing a proof if this result.