As indicated by the phrase "refraining from greetings" due to mourning, the concept of sending the postcard is to say, "We are sorry we can't send a New Year's greeting due to the mourning period," as opposed to, "We won't accept a New Year's greeting from you," (while some people believe it's rude to send the nengajo to a mourning family, it's common to send a kanchuomimai [winter greeting card] instead).
これは、喪中「欠礼」という言葉の示すとおり、「年賀の挨拶をお断りします」というよりは、「自分の家は今年は忌中なので年賀の挨拶ができなくて申し訳ありません」という意味だからである（昨今では喪中の家に年賀状を出すのは失礼という人もいる。一般的には寒中御見舞いのはがきを出すことが多い。）。