Three-dimensional hydrodynamic collapse calculations of elongated isothermal clouds with solid-body rotation about an arbitrary axis are presented. Four parameters are required to specify our models: the ratio of length to diameter of the cylinders, L/D, the initial Jeans number J0 (ratio of the absolute value of gravitational to thermal energies), and the ratio of the absolute value of the rotational to gravitational energies for the components of rotation parallel and perpendicular to the cloud's major axis, beta(parallel-to) and beta(perpendicular-to), respectively. Four different modes of fragmentation are identified. In all evolutions that formed more than one fragment, a structural fragmentation mode occurred first, forming condensations on each side of the cylinder. This mode is referred to as Binary Fragmentation. For low J0 and beta(parallel-to), the Binary Fragments evolve into a binary system with each fragment surrounded by a disk. These disks are parallel but not coplanar. At higher J0, the circumfragmentary disks fragment (Disk Fragmentation) due to their mutual gravitational interaction. With a higher value of beta(parallel-to), fragmentation occurs via an intermediary bar stage (Bar Fragmentation). In some cases, the bar fragments into one subcondensation and a spiral arm which subsequently fragments (Bar-Arm Fragmentation). The occurrence of these modes is located in the parameter space J0 and beta(parallel-to). The parameter beta(perpindicular-to) also has some effect on the outcome, but it does not determine the types of fragmentation modes. The multiple systems formed by these processes are not coplanar. Fragmentation processes considered in previous studies cannot explain the observation that at least one-third of multiple systems are non-coplanar. This was our main motivation for introducing rotation about arbitrary axis. The calculations show that gravitational torques and tidal effects control most of the fragmentation and coalescence which occur. The fragmentation processes and gravitational torques can also reduce the fragment's specific angular momentum by up to two orders of magnitude compared to that of the parent cloud. This is consistent with the inferred value of the specific angular momentum of the initial protosolar nebula. The importance of considering star formation on a large scale is clear from the presence of these new fragmentation modes. This gives an indication of the dynamical processes involved in the fragmentation of large molecular cloud complexes.