Sian Busby, who was diagnosed with lung cancer five years ago, died in September.
Mr Peston, who lives in north London, tweeted: "Burgled last night. My beloved late wife's 1920s engagement and wedding rings taken. Naive to think they might turn up, but..."
Ms Busby, a novelist, was a schoolfriend of Mr Peston's sister and the pair began dating when they were studying at different universities.
The couple have a teenage son, Max.