C++: Improving cache performance in a 3d array

I don't know how to optimize cache performance at a really low level, thinking about cache-line size or associativity. That's not something you can learn overnight. Considering my program will run on many different systems and architectures, I don't think it would be worth it anyway. But still, there are probably some steps I can take to reduce cache misses in general.

Here is a description of my problem:

I have a 3d array of integers, representing values at points in space, like [x][y][z]. Each dimension is the same size, so it's like a cube. From that I need to make another 3d array, where each value in this new array is a function of 7 parameters: the corresponding value in the original 3d array, plus the 6 indices that "touch" it in space. I'm not worried about the edges and corners of the cube for now.

Here is what I mean in C++ code:

void process3DArray (int input[LENGTH][LENGTH][LENGTH], 
                     int output[LENGTH][LENGTH][LENGTH])
{
    for(int i = 1; i < LENGTH-1; i++)
        for (int j = 1; j < LENGTH-1; j++)
            for (int k = 1; k < LENGTH-1; k++)
            //The for loops start at 1 and stop before LENGTH-1
            //or other-wise I'll get out-of-bounds errors
            //I'm not concerned with the edges and corners of the 
            //3d array "cube" at the moment.
            {
                int value = input[i][j][k];

                //I am expecting crazy cache misses here:
                int posX = input[i+1] [j]   [k];
                int negX = input[i-1] [j]   [k];
                int posY = input[i]   [j+1] [k];
                int negY = input[i]   [j-1] [k];
                int posZ = input[i]   [j]   [k+1];
                int negZ = input[i]   [j]   [k-1];

                output [i][j][k] = 
                    process(value, posX, negX, posY, negY, posZ, negZ);
            }
}

However, it seems like if LENGTH is large enough, I'll get