The governor of Mecca and the governor of the capital Riyadh were replaced as were several senior religious officials.
King Salman - who was a half-brother of the late King Abdullah - also gave a bonus of two months' salary to all Saudi state employees and military personnel. Pensioners and students received similar bonuses.
The changes were announced in 30 royal decrees.
"Dear people: You deserve more and whatever I do will not be able to give you what you deserve," the king said later on his Twitter account.
He asked citizens to "not forget me in your prayers".
The Saudi Press Agency said King Salman had relieved intelligence chief Prince Khalid bin Bandar bin Abdul Aziz al-Saud of his post.
Saudi King Abdullah speaks at his private residence in the Red Sea city of in Jeddah, Saudi Arabia. 27 June 2014 King Abdullah had been suffering from a lung infection before he died
General Khalid bin Ali bin Abdullah al-Humaidan becomes the new intelligence chief, with cabinet rank.
A nephew of the late king, Prince Bandar bin Sultan, was removed from his posts as secretary general of the National Security Council and adviser to the king.
Two of Abdullah's sons were also removed - Prince Mishaal, governor of the Mecca region, and Prince Turki, who governed the capital Riyadh.
The Oil Minister, Ali al-Naimi, Foreign Minister Prince Saud al-Faisal and Finance Minister Ibrahim al-Assaf remain in the cabinet.
Another of Abdullah's sons, Prince Mutaib, stays as minister of the National Guard.