As Allen (Sha Tin College) discovered, the trick to cracking this problem is to work systematically:

"My method was to start of with a small cube and work onwards from that. I started off with a $3 \times 3 \times 3$ cube and stared counting the number of lines for each Vertical, Horizontal and Diagonal"

Doing this he discovered that there are 13 winning lines that go through middle-middle-middle.

The winning lines in general seperate into three types:

"Lines" of three cubes can be made from cubes joined face to face

"Diagonals" are cubes joined edge to edge in a line - so the diagonal of a face for example

"Long Diagonals"are lines of three cubes joined vertex to vertex, going through the middle from a vertex to one diagonally opposite.

Counting these gives 27 lines, 18 diagonals and 4 long diagonals for the $3 \times 3 \times 3$ cube. In general, for an $n \times n \times n$ cube:

Lines: $3n^2$

Diagonals: $6n$

Long Diagonals: 4