The notion of dynamical entropy for actions of a countable free abelian group $G$ by automorphisms of $C^*$-algebras is studied. These results are applied to Bogoliubov actions of $G$ on the CAR-algebra. It is shown that the dynamical entropy of Bogoliubov actions is computed by a formula analogous to that found by Størmer and Voiculescu in the case $G={\bf Z}$, and also it is proved that the part of the action corresponding to a singular spectrum gives zero contribution to the entropy. The case of an infinite number of generators has some essential differences and requires new arguments.