Last year, Denis Chebikin introduced the alternating descent of a permutation. For a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $S_n$, let $AltDes(\sigma) = \{2i+1: \sigma_{2i+1} <\sigma_{2i+2}\} \cup \{2i: \sigma_{2i} > \sigma_{2i+1}\}$. Then we can define $altdes(\sigma) =|AltDes(\sigma)|$ and $altmaj(\sigma) =\sum_{i \in AltDes(\sigma)} i$. We shall show how to derive the generating function for the joint distribution of $altdes$ and $altmaj$ over the symmetric group. Slight variations of our technique also allow us to find similar generating functions for the groups $B_n$ and $D_n$.