The stochastic Allen-Cahn equations, as a typical example of nonlinear stochastic partial differential equations, play an important role in phase theory. In this paper, we investigate the rate of convergence in the $p$th mean for a truncated-type explicit Euler time-stepping method applied to the stochastic Allen-Cahn equations as well as using the spectral Galerkin approximation in spatial discretization. Finally, a numerical example is given to confirm the strong convergence order.