Traffic monitoring is a key enabler for several planning and management activities of a Smart City. However, traditional techniques are often not cost efficient, flexible, and scalable. This paper proposes an approach to traffic monitoring that does not rely on probe vehicles, nor requires vehicle localization through GPS. Conversely, it exploits just a limited number of cameras placed at road intersections to measure car end-to-end traveling times. We model the problem within the theoretical framework of network tomography, in order to infer the traveling times of all individual road segments in the road network. We specifically deal with the potential presence of noisy measurements, and the unpredictability of vehicles paths. Moreover, we address the issue of optimally placing the monitoring cameras in order to maximize coverage, while minimizing the inference error, and the overall cost. We provide extensive experimental assessment on the topology of downtown San Francisco, CA, USA, using real measurements obtained through the Google Maps APIs, and on realistic synthetic networks. Our approach provides a very low error in estimating the traveling times over 95% of all roads even when as few as 20% of road intersections are equipped with cameras.