It is difficult to give a straight answer to the question, because if we define peace as the absence of war (example), we still have the problem of the nebulous definition of war. Does war include guerilla wars, civil wars, insurgencies and other forms of conflict? Does war have a clearly defined start and end date?
For example, when did World War II end? While the Japanese ceased-fire and announced surrender in August 1945, signed the paperwork in September 1945, signed the peace treaty in 1951, and were no longer occupied by the US in 1952. Some argue that some countries never signed a peace treaty, but just stopped fighting, so are technically still at war. [Reference - not a strong source, but this isn't controversial.]
Notwithstanding that, I am taking advantage of the looseness of the quote in the question, which uses a very broad "some sort of war", to include Civil Wars and Rebellions. The following conflicts overlapped each other, leaving no days where the entire world was at peace.
These are, by no means, the only conflicts during that time, but I picked some of the longer conflicts to cover the entire period.