Monday's Medina attack, which killed four policemen, followed suicide bombings in Jeddah and eastern city of Qatif.
Saudi Arabia's King Salman bin Abdulaziz Al Saud has vowed that his government will "strike with an iron fist", a day after a series of bombings hit the country.
Four people were killed at the Prophet's Mosque in Medina, Islam's second holiest site, on Monday, as suicide attackers also struck two other cities.
"The kingdom is fully determined to strike with an iron fist all those who aim at the minds or ideas of our dear young people," Salman said on Tuesday in an address to the nation for the Islamic feast of Eid al-Fitr.
Monday's Medina attack, 24 hours before the end of the fasting month of Ramadan, has drawn condemnation from Muslim leaders worldwide.
The Al Saud ruling family considers itself the protectors of Islam's holiest sites, Medina and Mecca.
Following the attack, Mohammed bin Nayef bin Abdul-Aziz, Saudi's crown prince who is spearheading the country's anti-terror efforts, visited wounded victims, as he sought to reassure Saudis that the country's security "is at its highest levels".
"I know confronting terror operations is not simple. The simple repercussions you feel following the explosion will go away. I’ve been through this experience before and I [understand] how you feel," Al Arabiya TV quoted him as saying.
No group has claimed responsibility for the attack in Medina, or for two other suicide bombings the same day outside a Shia mosque in the eastern city of Qatif and near the US consulate in Jeddah.
The Jeddah bombing killed only the attacker, and no casualties other than the bomber have been reported in Qatif.
The Saudi interior ministry identified the Jeddah attacker as Abdullah Waqar Khan, a Pakistani national in his early 30s. In a tweet, the ministry said that Khan, a driver, had moved to Jeddah 12 years ago to live with his wife and her parents.
Pakistan said on Tuesday that it was going to investigate whether the suicide bomber in Jeddah was one of its nationals.
Many observers suspect the Islamic State of Iraq and the Levant (ISIL, also known as ISIS) of being behind the bombings.
The group, which controls areas of eastern Syria and northern Iraq, has frequently denounced the Saudi monarchy and has claimed previous attacks on Shia mosques in Qatif and elsewhere in the kingdom.
The group has claimed responsibility for Sunday's car bomb attack on a shopping street in the Iraqi capital Baghdad in which more than 200 people were killed.
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