Solution:

Choose $3$ of the $5$ rows in $\binom{5}{3}=10$ ways and $3$ of the $5$ columns in $\binom{5}{3}=10$ ways.

The three chosen blocks must occupy distinct rows and columns, so they form a matching between the three rows and three columns, which can be done in $3!=6$ ways.

The total is $10\cdot 10\cdot 6=600.$

Thus, the correct answer is C.