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There are rising scenarios in communication systems, where the noises exhibit impulsive behavior and are not adequate to be modeled as the Gaussian distribution. The generalized Gaussian distribution instead is an effective model to describe real-world systems with impulsive noises. In this paper, the problem of efficiently evaluating the error performance of linear block codes over an additive white generalized Gaussian noise (AWGGN) channel is considered. The Monte Carlo (MC) simulation is a widely used but inefficient performance evaluation method, especially in the low error probability regime. As a variance-reduction technique, importance sampling (IS) can significantly reduce the sample size needed for reliable estimation based on a well-designed IS distribution. By deriving the optimal IS distribution on the one-dimensional space mapped from the observation space, we present a general framework to designing IS estimators for memoryless continuous channels. Specifically, for the AWGGN channel, we propose an $L_p$-norm-based minimum-variance IS estimator. As an efficiency measure, the asymptotic IS gain of the proposed estimator is derived in a multiple integral form as the signal-to-noise ratio tends to infinity. Specifically, for the Laplace and Gaussian noises, the gains can be derived in a one-dimensional integral form, which makes the numerical calculation affordable. In addition, by limiting the use of the union bound to an optimized $L_1$-norm sphere, we derive the sphere bound for the additive white Laplace noise channel. Simulation results verify the accuracy of the derived IS gain in predicting the efficiency of the proposed IS estimator.