Tikrit was seized by IS in June 2014, but the city has largely been spared the attacks that have plagued other parts of the country since its recapture.
Militants shot dead four security personnel at a checkpoint on the west side of the city, then continued north and detonated an explosives-rigged vehicle at another checkpoint, killing eight more people, a police colonel and lieutenant colonel said.
The attacks also wounded a total of 23 people, according to the officers.
Hometown of executed dictator Saddam Hussein, Tikrit lies 60 kilometres north of Baghdad and was the second city to fall to IS in 2014 as the jihadists swept through large areas north and west of the capital.
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It was also the first city to be recaptured by Iraqi forces, which are now preparing for a push on Mosul, a city north of Tikrit that is the last one held by IS in the country.
Western officials have indicated that the drive for Mosul, Iraq's second city, may begin next month, though Baghdad has declined to specify when it will start.
Even after Mosul is retaken, the jihadists will still have the ability to carry out attacks like those in Tikrit on Saturday.
Analysts have warned that IS is likely to resort to more such attacks in the face of the loss of its last urban bastion in Iraq.