There are indications throughout the "A Song of Ice and Fire" series that the people in the North follow practices that have existed in Westeros for longer than the practices followed in the rest of Westeros. The Northern form of life is part of an unbroken tradition that started with "the First Men", who were the first human migrants to Westeros. The most obvious example of this is the religion of the North; Northerners follow "the Old Gods", as opposed to the religiion of the Seven, which was introduced by a group of people who migrated to Westeros later ("the Andals").
When Bran says "Our ways are the old ways", he is saying that we, in the North, follow the old practices of the First Men. Presumably, these old practices include the practice of the man who passes the sentence being the man who swings the sword.