In this paper, new exact and approximate moment generating functions (MGF) expression for generalized fading models are derived. Specifically, we consider the $η-λ-μ$, $α-μ$, $α-η-μ$, $α-λ-μ$, $α-κ-μ$, and $α-λ-η-μ$ generalized fading distributions to derive approximate MGF expressions. The new expressions are very accurate and, in contrast to earlier results in the literature, avoid any complicated special functions, e.g. the Meijer-$G$ and Fox $H$-functions. As such, the new MGF expressions allow easier and more efficient analytical manipulations, which also apply for their special cases such as the Rayleigh, Rice, and Nakagami-m fading. As an illustrative application, the average bit error rates for each of the fading models are evaluated using the new derived MGF expressions. The accuracy of the analytical results by using the numerically computed results as a basis of comparison as well as published results from the literature.