A new Monte Carlo (MC) algorithm, the 'dose planning method' (DPM), and its associated computer program for simulating the transport of electrons and photons in radiotherapy class problems employing primary electron beams, is presented. DPM is intended to be a high accuracy MC alternative to the current generation of treatment planning codes which rely on analytical algorithms based on an approximate solution of the photon/electron Boltzmann transport equation. For primary electron beams, DPM is capable of computing 3D dose distributions (in 1 mm3 voxels) which agree to within 1% in dose maximum with widely used and exhaustively benchmarked general-purpose public-domain MC codes in only a fraction of the CPU time. A representative problem, the simulation of 1 million 10 MeV electrons impinging upon a water phantom of 128(3) voxels of 1 mm on a side, can be performed by DPM in roughly 3 min on a modern desktop workstation. DPM achieves this performance by employing transport mechanics and electron multiple scattering distribution functions which have been derived to permit long transport steps (of the order of 5 mm) which can cross heterogeneity boundaries. The underlying algorithm is a 'mixed' class simulation scheme, with differential cross sections for hard inelastic collisions and bremsstrahlung events described in an approximate manner to simplify their sampling. The continuous energy loss approximation is employed for energy losses below some predefined thresholds, and photon transport (including Compton, photoelectric absorption and pair production) is simulated in an analogue manner. The delta-scattering method (Woodcock tracking) is adopted to minimize the computational costs of transporting photons across voxels.