Born in the Soviet Republic of Georgia, Jordania studied at the Tblisi Conservatory and the Leningrad Conservatory. In 1971, he won the Herbert von Karajan competition in Berlin, then served as assistant conductor at the Leningrad Philharmonic. Over the next decade, he guest-conducted leading Soviet and Eastern European orchestras and led Ukraine's Kharkov Philharmonic.
According to the Post, however, Jordania was frustrated by his isolation from Western composers and performers. In 1983, he and violinist Viktoria Mullova, then his girlfriend, defected to the West during a tour of Finland (Jordania left his wife and children behind).
Jordania settled in the United States, where he appeared frequently as a guest conductor. He was music director of the Chattanoga Symphony and Opera from 1985 to 1992 and of the Spokane Symphony from 1991 to 1993.
After the fall of the Soviet Union, Jordania was named the Kharkov Philharmonic's laureate chief conductor for life, and the orchestra created a conducting competition in his name. He also held positions at the Korean Broadcasting System Symphony Orchestra in Seoul, the Russian Federal Orchestra of Moscow, and South Korea's Daegu City Symphony Orchestra.