A class of multidimensional designs is presented for factorial experiments with two treatment factors and two or more orthogonal blocking factors. We assume that interactions between all factors, except for the two treatment factors, are negligible. A table of efficient three-dimensional designs is given together with tables of component designs that can be used to construct two-, three-, or higher-dimensional designs. All the multidimensional designs considered in this article haved> 2 crossed orthogonal blocking factors and exactly one treatment combination observed at each combination of levels of the blocking factors. They are constructed by amalgamating one-dimensional generalized cyclic designs with known properties. The purpose of this article is to increase the availability of multidimensional designs, three-dimensional designs in particular. Our main objective is to achieve efficient estimation of the factorial effects via least squares estimators that are uncorrelated after adjustment for fixed-block effects; that is, we require designs withorthogonal factorial structure. This property is guaranteed for thed-dimensional designs considered in this article. Several tables of designs are given. The first lists a selection of efficient three-dimensional designs together with their generalized cyclic components and their efficiency factors. The remaining tables list generalized cyclic block designs that can be used either as (one-dimensional) block designs or chosen for amalgamation into a multidimensional design. The multidimensional designs can be constructed so that they possess the property ofadjusted orthogonalityof the block effects after adjusting for the treatment effects. This property allows its factorial efficiency factors to be calculated simply from the factorial efficiency factors of its component designs.