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:
Long Diagonals: 4