The method of characteristics (MOC) in unstructured meshes has become a standard for reactor physics applications. One of the major drawbacks of the MOC is the difficulty to implement higher-order integration schemes to improve spatial convergence. In this paper we present a high-order MOC spatial discretization that uses linear interpolation on surface values for the collision source. This conservative linear surface (CLS) scheme exhibits parabolic convergence with the mesh size but lacks positivity. Numerical results for the well-known Stepanek benchmark and for more realistic boiling water reactor assemblies show CLS faster convergence over the standard step characteristics scheme. A generalization of the synthetic DPN acceleration scheme provides an efficient method to accelerate the internal transport iterations.