There are three basic vampyres that I have found in Sumerian myths - the Ekimmu, the Uruku (Utukku), and the Seven Demons.

The Ekimmu: This creature was said to be created when death happened violently or the burial was not handled properly. Author Konstantinos believes that these creatures were intentional psychic vampyres. These creature were astral in nature, they hunted for a victim they could "seize and torment." The person could only be released by an exorcism performed by a priest or priestess.

The Uruku (Utukku): This creature is referred to as a "vampyre which attacks man" in a cuniform inscription. I am attempting to locate more information on this reference which also came from Konstantinos.

The Seven Demons: These demons are mentioned in many texts and incantations of the Mesopotamian cultures. One Sumerian banishing rite describes them as immortal blood drinkers.