This is my proposition, I know it does not looks super sexy, but I believe it's faster (3X faster for the sample string) and requires the exact amount of memory than all the ones using Select, Join, and all that jazz :-)

static string ConvertString(string s)
{
    char[] newS = new char[s.Length * 2 + 1];
    int i = 0;
    do
    {
        newS[i] = s[i / 2];
        if (i == (s.Length * 2 - 2))
            break;

        i++;
        newS[i] = '.';
        i++;
    }
    while (true);
    return new string(newS);
}

Plus it does not require the Framework 4.