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The dynamic response of three‐dimensional rigid surface foundations of arbitrary shape is numerically obtained. The foundations are placed on a linear elastic, isotropic and homogeneous half‐space representing the soil medium and are subjected to either external dynamic forces or seismic waves of various kinds and directions, with a general transient time variation. The problem is formulated in the time domain by the boundary element method and the response is obtained by a time step‐by‐step integration. Two examples dealing with three‐dimensional rectangular foundations are presented in detail, together with comparisons with other methods, in order to document the accuracy of the method. The main advantages of the proposed method are that, unlike frequency domain techniques, it provides directly the transient response and forms the basis for extension to the case of non‐linear behaviour.