RSA signature attack check on lowest 16 bytes - Implementation
if(checkValue.compareTo(resultOfModulus) == 0)


Here's your problem; you're checking if $y^3 = (md5BigInteger \bmod 2^i)$; that's wrong (as $md5BigInteger \bmod 2^i$ won't typically be a cube).

Instead, what you want is to check if $y^3 \equiv md5BigInteger \pmod{2^i}$; or, in other words (thinking less like a mathematician, and more like a programmer) if $(y^3 \bmod 2^i) = (md5BigInteger \bmod 2^i)$

Also, don't forget to check if $md5BigInteger$ is odd (or more generally, $8^k w$ for integer $k$ and odd $w$); if it's not of that form, no such simple cube will exist.