I don't know about knowing the specific date (June 1, 1307), but they would have known the date in terms of the church calendar at the very least. This would extend beyond simply designating the Sabbath as the beginning/end of each week, but also many feast days would fall in the middle of the week due to their more specific nature, such as St. Valentine's day on February 14. In this case, they would have most likely relied upon the local clergy to inform them of the upcoming date.

Also, months can be easily determined by observing the lunar cycle, which was important anyways for the farming seasons.

hi! a few related posts you may find interesting

on the church calendar / feast days

• How much free time did an average person in the middle ages have and how die he/she spend it? - re the frequency of feast days

• In medieval England, how "mandatory" was it for everyone to attend church services? Were there punishments for people caught skipping church? - re expected church attendance

• What were important every-village-has-one buildings in the Medieval ages? - more on expected church attendance

on knowing the year

• This is probably a dumb question, but did your average peasant in the dark ages know what year it was?

• Would a normal person in the Middle Ages have known what year they were living in?

The Jewish calendar shows how it worked pretty effectively. There are holidays and festivals based on the day of the lunar month, days based on shabbats (which are differentiated by each having its own torah and haftorah portion), and a lot of holidays based off of counting days from a major holiday, complete with a slightly different prayer each day.