To find the unit rate of books read per month, you need to divide the total number of books read by the number of months in a year.
Liam read a total of 36 books in 12 months.
So, the unit rate is:
\[ \text{Unit rate} = \frac{36 \text{ books}}{12 \text{ months}} = 3 \text{ books per month} \]
Therefore, Liam read 3 books per month.