  Motivated by Murre's work on universal regular homomorphisms on Chow groups
in codimension $2,$ we generalize the algebraic equivalence relation and
regular homomorphisms to the context of Voevodsky motives over a field. In the
Nisnevich topology, we prove the existence of \emph{universal} regular
homomorphisms for a certain class of motivic cohomology groups, recovering
Murre's theorem and the existence of Picard and Albanese varieties as special
cases. This class also includes interesting cases such as higher Chow groups
and Milnor $K$-groups. The appendix by Kahn proves that, for \'etale motives,
universal regular homomorphisms exist for all geometric motives and compares
them with those in the Nisnevich topology when both exist.
