While it wouldn't exactly follow the lunar cycle as the parent mentions, there are already proposed calendars similar to what you mention[1]. Of course, with any calendar readjustment come just as many drawbacks, as the wiki entry mentions some.
[1]https://en.wikipedia.org/wiki/International_Fixed_Calendar
I'm disappointed that this wiki page mentions no drawbacks to my plan to slow down the rotation of the earth. I've no idea what they would be, but I'm guessing that this might come with some negative repercussions.
What we really need to do is use some rockets to slightly slow down the rotation of the earth so that a day is exactly 1/360th of a year. All kinds of numbers divide evenly into 360 so that should make building a good calendar easy.