  We study the analytic Bethe ansatz in solvable vertex models associated with
the Yangian $Y(X_r)$ or its quantum affine analogue $U_q(X^{(1)}_r)$ for $X_r =
B_r, C_r$ and $D_r$. Eigenvalue formulas are proposed for the transfer matrices
related to all the fundamental representations of $Y(X_r)$. Under the Bethe
ansatz equation, we explicitly prove that they are pole-free, a crucial
property in the ansatz. Conjectures are also given on higher representation
cases by applying the $T$-system, the transfer matrix functional relations
proposed recently. The eigenvalues are neatly described in terms of Yangian
analogues of the semi-standard Young tableaux.
