An extensive network of dose rate monitoring stations continuously measures ambient dose rate across Europe, as part of the EURDEP system. Its purpose is early warning in radiological emergencies and documenting its temporal and spatial evolution. In normal conditions, when there is no contribution to the dose rate signal coming from fresh anthropogenic contamination, the data represent the radiation "background", i.e. the combined natural radiation and existing anthropogenic contamination (by global and Chernobyl fallout). These data are being stored, but have so far not been evaluated in depth, or used for any purpose. In the framework of the EU project 'European Atlas of Natural Radiation' the idea has emerged to exploit these data for generating a map of natural terrestrial gamma radiation. This component contributes to the total radiation exposure and knowing its geographical distribution can help establishing local 'radiation budgets'. A further use could be found in terrestrial dose rate as a proxy of the geogenic radon potential, as both quantities are related by partly the same source, namely uranium content of the ground. In this paper, we describe in detail the composition of the ambient dose equivalent rate as measured by the EURDEP monitors with respect to its physical nature and to its sources in the environment. We propose and compare methods to recover the terrestrial component from the gross signal. This requires detailed knowledge of detector response. We consider the probes used in the Austrian, Belgian and German dose rate networks, which are the respective national networks supplying data to EURDEP. It will be shown that although considerable progress has been made in understanding the dose rate signals, there is still space for improvement in terms of modelling and model parameters. An indispensable condition for success of the endeavour to establish a Europe-wide map of terrestrial dose rate background is progress in harmonising the European dose rate monitoring network.