The lattice Boltzmann method is a highly promising approach to the simulation of complex flows. Here, we realize recently proposed multispeed lattice Boltzmann models [S. Chikatamarla, Phys. Rev. Lett. 97 190601 (2006)] by exploiting the flexibility offered by off-lattice Boltzmann methods. The approach is based on the general characteristic-based algorithm for off-lattice Boltzmann simulations that preserves all appealing properties of the standard lattice Boltzmann method while extending the method to unstructured grids. We show that the use of multispeed models indeed gives rise to major improvements in accuracy. The suggested approach thus renders truly large-scale off-lattice Boltzmann computations practical.