How do you want the 3D array to be displayed? There's no way around iterating through all elements. A 3D array is difficult to view in a reasonable way. One would be to treat it like it was a jagged array, so you have 3 nested arrays. Something like that:

[[[1,2,3],[4,5,6]],[[7,8,9],[10,11,12]]]

this is not very helpful if there are many values since the structure is really hard to read.

What data is stores in the array? And how many elements per dimension do you have? Just to clarifiy, my example above has an element counr of 2, 2, 3 which is not that much.

A function that forms a string like that could look like this:

// C#
string Format3DArray(int[,,] array)
{
    string result = "";
    for(int d0 = 0; d0 < array.GetLength(0);d0++)
    {
        if (d0>0) result += ",";
        result += "[";
        for(int d1 = 0; d1 < array.GetLength(1);d1++)
        {
            if (d1>0) result += ",";
            result += "[";
            for(int d2 = 0; d2 < array.GetLength(2);d2++)
            {
                if (d2>0) result += ",";
                result += array[d0,d1,d2].ToString();
            }
            result += "]";
        }
        result += "]";
    }
    result += "]";
    return result;
}

Usually it's better to use a StringBuilder for this, but since it's just for debugging i don't care about performance ;)

ps. i just wrote that from scratch so forgive any potential typing / syntax errors ;)

edit
You can also use MonoDevelop to debug your game and insert a breakpoint so you can inspect the variables directly. Never used multidim arrays when i was debugging, so i don't know how MonoDevelop will display such an array.

I may be misunderstanding you but it looks like you need to override ToString() in your Vector3i object.