Solution:

Suppose that one corner of the rectangle is on $(20,19)$. First of all, note that $20$ and $19$ are relatively prime. This means that the line does not intersect any vertex of a unit square in the interior of the grid.

Now, consider the diagonal from $(0,0)$ to $(20,19)$. This diagonal intersects exactly $19$ vertical segments and $18$ horizontal segments. Here, each intersection point represents an entry point into a new unit square. Including the original unit square (which has the origin as a corner), this accounts for a total of $19+18+1=38$ unit squares.

Similarly, the other diagonal of the rectangle passes through $38$ unit squares. Finally, since the diagonals intersect at the center, the center must have been an entry point for one diagonal and an exit point for the other, and vice versa. These pertain to the two unit squares that both diagonals pass through.

Therefore, the diagonals pass through a total of $2(38)-2=74$ unit squares.