Summary: Direct numerical simulation (DNS) of a spatially developing turbulent boundary layer (TBL) over a wall roughened with regularly arrayed cubes was performed to investigate the effects of three-dimensional (3-D) surface elements on the properties of the TBL. The cubes were staggered in the downstream direction and periodically arranged in the streamwise and spanwise directions with pitches of \(p_{x}/k = 8\) and \(p_{z}/k = 2\), where \(p_{x}\) and \(p_{z}\) are the streamwise and spanwise spacings of the cubes and \(k\) is the roughness height. The Reynolds number based on the momentum thickness was varied in the range \(Re_{\theta } = 300 - 1300\), and the roughness height was \(k = 1.5\theta_{in}\), where \(\theta_{in}\) is the momentum thickness at the inlet, which corresponds to \(k/\delta = 0.052-0.174\) from the inlet to the outlet; \(\delta\) is the boundary layer thickness. The characteristics of the TBL over the 3-D cube-roughened wall were compared with the results from a DNS of the TBL over a two-dimensional (2-D) rod-roughened wall. The introduction of cube roughness affected the turbulent Reynolds stresses not only in the roughness sublayer but also in the outer layer. The present instantaneous flow field and linear stochastic estimations of the conditional averaging showed that the streaky structures in the near-wall region and the low-momentum regions and hairpin packets in the outer layer are dominant features in the TBLs over the 2-D and 3-D rough walls and that these features are significantly affected by the surface roughness throughout the entire boundary layer. In the outer layer, however, it was shown that the large-scale structures over the 2-D and 3-D roughened walls have similar characteristics, which indicates that the dimensional difference between the surfaces with 2-D and 3-D roughness has a negligible effect on the turbulence statistics and coherent structures of the TBLs.