It is well known that the Belousov-Zhabotinskii reaction can exhibit planar traveling waves in the excitable-kinetics regime. Using the previously formulated piecewise-linear Oregonator model, we study the stability of these waves to general disturbances. We find possible linear instabilities, either to transverse structure or to a longitudinal oscillation; these set in below a critical velocity, and which one occurs first depends on the ratio of the diffusivities of the two chemical species. We also consider the weakly nonlinear bifurcation analysis for the steady-state bifurcation, showing that the bifurcation is supercritical over a wide parameter range. The implications of these results for experimental systems are briefly discussed.