I have done a complete search on paper for a complete
checkboard. It cannot be done for any cobe of even side. The
restrictions are in the corners. I would like somebody to double
check this though. The standard problems to run into are:
You need two of one type of corner, or
You have to rotate exactly one corner, which is impossible.

There are a couple other amazing things I found.
As it is known, any two edges can be exchanged (or appear to be
exchaged because of center arbitrariness). It is ALSO possible
to (appear to) exchange two corners, for about the same reason.
This is impossible on the 3^3 becuase it requires the exchange of
two edges. But in the 4^3 there are two cubies per 3^3 edge.
Therefore, we just do a double exchange, which does not violate
any parity arguments. Combining these two moves, you can flip a
1x1x4 edge.

Happy revenge.