High-resolution neutron scattering experiments on optimally doped La2-xSrxCuO4 (x=0.16) reveal that the magnetic excitations are dispersive. The dispersion is the same as in YBa2Cu3O6.85, and is quantitatively related to that observed with charge sensitive probes. The associated velocity in La2-xSrxCuO4 is only weakly dependent on doping with a value close to the spin-wave velocity of the insulating (x=0) parent compound. In contrast with the insulator, the excitations broaden rapidly with increasing energy, forming a continuum at higher energy and bear a remarkable resemblance to multiparticle excitations observed in 1D S=1/2 antiferromagnets. The magnetic correlations are 2D, and so rule out the simplest scenarios where the copper oxide planes are subdivided into weakly interacting 1D magnets.