If you think of yourself, or if others perceive you, as being of Italian origin, when you are in Italy they will consider you oriundo.
Italians place a high value on "blood" (ethnic identity) or jus sanguinis. So you may well be considered oriundo even if only one parent was born there, or even one grandparent, especially if it was your paternal grandfather- so that you have the Italian surname.
Il grande giocatore di baseball Joe DiMaggio era oriundo. Era nato in California, ma i suoi genitori erano siciliani.
The great baseball player Joe DiMaggio was oriundo. He was born in California, but his parents were Sicilian.
La famosa cantante americana Madonna e' oriunda.
The famous American singer Madonna is oriunda.
Tanti argentini sono oriundi.
Many Argentinians are of Italian origin.