The combination of an exact stochastic decomposition of non-Markovian dissipative quantum dynamics with the semiclassical initial value formalism is applied to Brownian motion in a Morse potential. The unified sampling of the stochastic noise and the semiclassical phase space distribution introduced in Koch et al. [W. Koch, F. Grossmann, J.T. Stockburger, J. Ankerhold, Non-Markovian semiclassical dynamics, Phys. Rev. Lett. 100 (2008) 230402] is laid out here in detail. By comparing our numerical results to those obtained by using the Caldeira-Leggett master equation, we show that even in the challenging regime of moderate friction and at low temperatures, where reservoir fluctuations are clearly non-Markovian, this approach allows for the accurate description of dissipative dynamics over many oscillation periods until thermalization is reached. (C) 2009 Elsevier B.V. All rights reserved.