The Hodge conjecture, may be neatly reformulated using motives: it holds iff the Hodge realization mapping any pure motive with rational coefficients (over a subfield k of C) to its Hodge structure is a full functor H : M(k)Q → HSQ (rational Hodge structures).
Similarly, the Tate conjecture is equivalent to: the so-called Tate realization, i.e. ℓ-adic cohomology is a faithful functor H: M(k)Qℓ → Repℓ(Gal(k)) (pure motives up to homological equivalence, continuous representations of the absolute Galois group of the base field k), which takes values in semi-simple representations.