A new method is proposed for deriving kinematically admissible velocity fields (KAVFs) for three-dimensional upper bound limit analyses in a Tresca material using coordinate transformations. The method allows the incompressibility condition to be satisfied simply by imposing certain requirements on the analytical form of velocity magnitudes. This allows for new classes of velocity fields to be derived solely using standard procedures. These new classes of fields include: KAVFs with new streamline shapes; new planar but non-plane-strain KAVFs; new radial but nonaxisymmetric KAVFs. The method allows the expression of local dissipation of plastic work in any field to be derived in a closed form. The proposed method makes an attempt to expand the applicability of three-dimensional upper bound limit analysis by introducing more realistic shapes of KAVFs, while maintaining simplicity and clear engineering meaning.