(Abridged) We study the gravitational radiation from gravitational collapses of rapidly rotating neutron stars induced by a phase-transition from normal nuclear matter to a mixed phase of quark and nuclear matter in the core of the stars. The study is based on self-consistent three dimensional hydrodynamic simulations with Newtonian gravity and a high resolution shock capturing scheme, and the quadrupole formula of gravitational radiation. The quark matter of the mixed phase is described by the MIT bag model and the normal nuclear matter is described by an ideal fluid equation of state (EOS). 1. We determined the magnitudes of the emitted gravitational waves for several collapse scenarios. 2. We determined the types and frequencies of the fluid oscillation modes excited by the process. In particular, we find that the gravitational wave signals produced by the collapses are dominated by the fundamental quadrupole and quasi-radial modes of the final equilibrium configurations. In some collapse scenarios, we find that the oscillations are damped out within a few dynamical timescales due to the growth of differential rotations and the formation of strong shock waves. 3. We showed that the spectrum of the gravitational wave signals is sensitive to the EOS, implying that the detection of such gravitational waves could provide useful constraints on the EOS of newly born quark stars. 4. For the range of rotation periods we have studied, we found no sign of the development of nonaxisymmetric dynamical instabilities in the collapse process.