We study a numerical scheme for the approximation of parabolic boundary-value problems with nonsmooth boundary data. This fully discrete scheme requires no boundary constraints on the approximating elements. Our principal result is the derivation of optimal convergence estimates in L(p)[O, T; L(2)(Omega)] norms for boundary data in L(p)[O, T; L(2)(Gamma)], 1 less than or equal to p less than or equal to infinity. For the same algorithms, we also show that the convergence remains optimal even in higher norms. The techniques employed are based on the theory of analytic semigroups combined with singular integrals.