One could be marking some of the cells and write down the order of visit of that cells.

Number of horizontal steps in each row (or in a set of marked rows if the grid is large)? In this example, 1012.

Or, as Rajesh suggests, put tiny labels A,B,C,D,... in the corners of some cells and ask to give the letters in the order they are encountered.

Rajesh's idea is probably nicer since entering the answer key involves checking one's solution to the puzzle, even though it overloads the grid with extraneous notations.

Denis