The first generation of inter-vehicle communication networks will most likely be based on the IEEE 802.11p standard. That is, they are going to deploy Carrier Sense Multiple Access (CSMA) to coordinate channel access among neighboring vehicles. Recently, concerns have been raised that fast-fading propagation conditions, i.e. a time- and frequency-selective fading as reported by several measurement campaigns in highway environments, might challenge the effectiveness of CSMA. These concerns also lead to the situation that alternative medium access control solutions are being discussed in standardization bodies. In this paper, we evaluate whether these concerns are justified or not. In comparison to previous studies, we use a high fidelity network simulator to study the extent by which the effectiveness of CSMA is reduced if such fading propagation conditions are considered. We also resolve the two reasons that may cause incoordination - either simultaneous transmission times or hidden terminal situations - and conclude that CSMA is able to effectively coordinate multiple access in vehicular radio channels as long as the load offered to the channel does not approach the maximum capacity.