The Hodge conjecture, with rational coefficients, states that for a complex smooth projective variety, the image of the cycle class map from Chow groups to Betti cohomology is the group of Hodge classes. Altough there are counter-examples for its integral version, Rosenschon and Srinivas proved that the étale version of the integral Hodge conjecture, i.e. using étale motivic cohomology groups instead of Chow groups, is equivalent to the Hodge conjecture with rational coefficients. In this talk, I will give an overview of trhe link between étale motivic cohomology and the Hodge conjecture, followed by a revisit to some of the counter-examples to the integral Hodge conjecture, such as the ones of Atiyah-Hirzebruch, Kollar and Benoist-Ottem, but from an étale motivic point of view.