There was no set race, but elections in ancient Greece required citizenship. Citizenship could only be attained by male members of society, so therefore only men could vote. City-states defined their citizenry differently, but the Athenian model is the model of democracy today.
To be an Athenian citizen you had to be free (not a slave) and originally you had to own property. This was later amended to include any male born of two Athenian parents. The citizen must also not have a personal or inherited mark against them. Personal marks were earned by violation of law or dishonor. Inherited marks were carried throughout generations and could only be removed by the vote of the assembly.
Although there was no set race, the limitations on citizenship fairly ensured only Greeks could vote. Currently citizenship is still required to vote, but the ways of becoming a citizen are more numerous and thus more forgiving of racial heritage.
In ancient Greece you had to be considered a citizen in order to vote, children and slaves were not considered citizens. In the older times women were considered to be citizens but not allowed to vote.