I develop a new wavefront-propagation method for calculating the traveltimes and geometrical amplitudes of seismic waves in a two-dimensional velocity model. The method assumes that the given velocity field is specified on a grid formed by regular triangular cells and that the velocity within each triangular cell varies linearly. These assumptions lead to a continuous representation of the velocity field. Five attributes are assigned to the wavefront at each grid point. They are traveltime, traveltime gradient, take-off angle, curvature radius of wavefront and geometric spreading factor. The attributes of a wavefront at one corner of a triangular cell are calculated from the attributes of wavefronts at other two corners by propagating the local circular wavefront through the triangular cell under the guidance of local rays. Starting from a given source position, these local computations proceed on an expanding half hexagonal ring. The new method computes the traveltimes of multiple arrivals by disassembling a multi-valued traveltime field into several single-valued fields, and then computing each of them separately. The method is evaluated with several synthetic velocity models, and its accuracy is checked with both analytical solutions and wave equation modeling.