IS seized Tikrit in June 2014, but the city has largely been spared the attacks that have plagued other parts of the country since its recapture last year.
Police officers said that 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.
The attacks also wounded a total of 23 people, according to the officers.
IS issued a statement claiming the attacks, but gave a different account of how they unfolded, saying that four militants — three Iraqis and a Tajik — were involved and detonated multiple bombs.
According to IS, the militants first attacked a checkpoint in the area with silenced weapons, killing its guards and leaving one jihadist behind to block the road.
One suicide bomber then set off a truck bomb at an entrance to the city, while two more detonated explosive vests at a gathering of police and pro-government paramilitaries close to a military base near Tikrit, the jihadist group said.
Hometown of executed dictator Saddam Hussein, Tikrit lies 60 kilometres (100 miles) 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.
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, although Baghdad has declined to specify when it will start.
While the announcement of the operation may be coming soon, Iraqi forces, which in some cases are dozens of kilometres (miles) away, will then have to fight their way to the city and surround it before an assault can take place.
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.