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Bit-interleaved coded modulation (BICM) has established itself as a quasi-standard for bandwidth- and power-efficient wireless communication. In this paper, we present an analytical approach to evaluate the performance of BICM transmission over frequency-flat fading additive white Gaussian noise channels. The statistic of the fading envelope is modeled as Nakagami-m distributed, which spans a wide range of practical multipath fading scenarios through adjustment of the m-parameter. For this setup, we derive approximations for the bit-error rate (BER) and cutoff rate of BICM. Different from previously proposed methods, our analysis is valid for general quadrature amplitude modulation and phase shift keying signal constellations and arbitrary bit-to-symbol mapping rules, and it results in simple closed-form expressions. The key idea is to use well-chosen subsets of signal points to approximate the probability density function of reliability metrics used for decoding. This approximation is accurate for signal-to-noise ratio regions of interest for BICM systems with moderate coding complexity such as, e.g., convolutional coded BICM systems. Based on this approximation we also derive an asymptotic BER expression, which reveals the diversity order and coding gain of BICM. The usefulness of the proposed analytical approach is validated through numerical and simulation results for a number of BICM transmission examples.