Let $F$ be a non-archimedean local field of residual characteristic different from $2$, and let $G$ be a unitary, symplectic or orthogonal group, considered as the fixed point subgroup in $\tilde G={\rm GL}(N,F)$ of an involution $\sigma$. We generalize the notion of a {\it simple character} for $\tilde G$, which was introduced by Bushnell and Kutzko [Annals of Mathematics Studies 129 (Princeton University Press, 1993)], to define semisimple characters. Given a semisimple character $\theta$ for $\tilde G$ fixed by $\sigma$, we transfer it to a character $\theta_{-}$ for $G$ and calculate its intertwining. If the torus associated to $\theta_{-}$ is maximal compact, we obtain supercuspidal representations of $G$, which are new if the torus is split only over a wildly ramified extension. 2000 Mathematics Subject Classification: 22E50.