An unsteady lifting-line theory for time-domain problems with arbitrary kinematics is presented. This is formulated by matching a vortex particle based 2D inner model with a 3D vortex lattice wake. This and a small-amplitude frequency-domain unsteady lifting-line theory are then verified against experiment and computational fluid dynamics for the case of a flat rectangular plate oscillating in heave at aspect ratios 3 and 6. Both lifting-line theories were found to generally be in good agreement with experimental and CFD results, providing reasonable solutions even in cases dominated by LEV shedding. For larger amplitude problems, the new non-linear geometry lifting-line theory predicted larger amplitudes and higher average lift coefficient than the small-amplitude lifting-line theory.