Easily constructible and optimally convergent generalisations of B-splines to unstructured meshes are essential for the application of isogeometric analysis to domains with non-trivial topologies. However, constructing smooth and optimally convergent isogeometric analysis basis functions, particularly for hexahedral meshes, remains an open question. We introduce a simple partition of unity construction that yields smooth blended B-splines, referred to as SB-splines, on unstructured quadrilateral and hexahedral meshes. To achieve this, we first define the mixed smoothness B-splines, which are $C^0$ continuous in the unstructured regions of the mesh but have higher smoothness everywhere else. Subsequently, the SB-splines are obtained by smoothly blending in the physical space the mixed smoothness B-splines with Bernstein bases of equal degree. A key novelty of our approach is that the required smooth weight functions are assembled from the available smooth B-splines on the unstructured mesh. The SB-splines are globally smooth, nonnegative, have no breakpoints within the elements and reduce to conventional B-splines away from the unstructured regions of the mesh. We demonstrate the excellent performance of SB-splines studying Poisson and biharmonic problems on semi-structured quadrilateral and hexahedral meshes, and numerically establishing their optimal convergence in one and two dimensions.