We consider the general motion (translation and rotation) of a deformable or rigid body of arbitrary shape in a linear shear flow of an effectively inviscid and incompressible fluid possessing uniform vorticity. The ambient vorticity may be time-dependent. For two-dimensional configurations a solution with uniform vorticity is possible for all times and for three-dimensional, it is possible only initially or during a short time interval after the body is impulsively introduced into the fluid. General analytic expressions for the vortical force and moment exerted on an arbitrary moving body are presented. Bearing in mind applications for large non-spherical bubble dynamics, the general expressions for the hydrodynamic loads are further reduced for symmetric quadratic shapes such as two-dimensional ellipses or three-dimensional ellipsoids. The simplified expressions are given in terms of the body's added-mass tensor, its six velocities and the ambient vorticity. The few available degenerate solutions for cylinders and spheres are readily obtained as limiting cases.