A quasisemiclassical trajectory method (QSCT) is proposed, which provides a practical procedure to study tunneling chemical reaction dynamics in multidimensional systems. QSCT incorporates the semiclassical tunneling paths that are generated by our previously proposed method [K. Takatsuka and H. Ushiyama, Phys. Rev. A 51, 4353 (1995)] into the so-called quasiclassical trajectory method, whereby the chemical reactions in a wide energy range are calculated in a systematic way. The accuracy of this approach is tested with the system of collinear H+H-H reaction on the so-called LSTH (Liu-Siegbahn-Truhlar-Horowitz) potential surface. The thermal rate constants at 1-3000 It are calculated. The resultant reaction rates are compared with the quantum mechanical values [Bondi et al., J. Chem. Phys. 76, 4986 (1982)], showing that they are in a systematically good agreement in this wide temperature range. We have also examined the dependence of the reaction probability on the initial sampling of the quasiclassical method. The contribution of the paths of dynamical tunneling to chemical reaction above the reaction threshold is estimated for the first time.