The country has also steamrollered fruit and burnt a huge pile of bacon.
The actions come a year after Russia banned some Western food products in retaliation to EU and US sanctions applied after Moscow annexed Crimea.
The destruction has caused an outcry from anti-poverty campaigners who say it should have been given to the poor.
One steamroller took an hour to crush nine tonnes of cheese. Another consignment was due to be burnt. Boxes of bacon have been incinerated. Peaches and tomatoes were also due to be crushed by tractors.
Religious leaders expressed outrage. One called the actions "insane, stupid and vile".
Russia has suffered notorious famines in its recent history which saw millions starve.
More than 285,000 people have backed an online petition calling on President Putin to give the food away.
The petition says that food sanctions have led to higher prices and shortages that are causing real hardship.
Food price inflation is running at around 20%.
Former prime minister, Mikhail Kasyanov, said that 20 million Russian citizens were below the poverty line, commenting that destroying food products was "some real triumph of humanism".
The EU is providing aid for European exporters of dairy produce, fruit and vegetables to ease the impact of the Russian ban on those goods.
The EU Commission said last week the "safety net" for Europe's dairy sector would remain in place until March 2016 and for fruit and vegetable growers until July 2016.
There have been daily road blockades by French farmers - and on Thursday similar protests erupted in Belgium. They want higher prices for their produce, saying they are struggling to stay in business.
