  In $(2n+1)$-dimensional non-Sasakian contact metric manifolds, we consider
Legendre curves whose mean curvature vector fields are $\mathcal{C}$-parallel
or $\mathcal{C}$-proper in the tangent or normal bundles. We obtain the
curvature characterizations of these curves. Moreover, we give some examples of
these kinds of curves which satisfy the conditions of our results.
