"Targets are plenty ... Start by hitting and destroying oil pipelines, factories and facilities which are the source (of income) of the tyrant government," the spokesman, Abu Hamza al-Muhajir, said in a recorded speech on the militant group's official telegram channel.
He said the kingdom had supported normalization with Israel by opening its airspace for Israeli flights to neighboring Gulf states.
The threats surfaced as Bahrain has followed the United Arab Emirates in agreeing last month to normalize ties with Israel.
Saudi Arabia, which has Islam's holiest sites and is the world's largest oil exporter, has stressed the need to step up efforts to reach a lasting and sustainable peace agreement among the Palestinians and the Israelis.