3-D Tic-Tac-Toe is one of those edge conditions in which apparently increasing the complexity of a game serves to make it unplayably trivial. Regular tic-tac-toe has a well-deserved reputation for being suitable only for children and idiots, as skilled players aware of the corner force will always draw, but 3D tic-tac-toe will always go to the first player. It remains true (although marginally less trivial) even on a board where the center is permanently blocked for both contestants. If the first player is barred from playing the center on his or her first turn, but the second player is under no such restrictions, then the second player has the winning advantage.
Traditional tic-tac-toe is broken because it is nearly impossible to create a fork: two simultaneous threats that cannot both be addressed by the defending player. Three-up 3D tic-tac-toe is broken for the opposite reason: the first player has sufficient control of the game that only the first move of the second player is not completely forced, and this single free play- unlike in traditional tic-tac-toe- is insufficient to prevent a force.
Both the "center-permitted" and "center-banned" strategies center around the same strategy: a sequence of moves played to form a fork, where every individual move past the first produces an immediate threat the second player must respond to with a specific play or lose immediately. This dominant play by the first player allows him or her to build any number of fork positions without meaningful interference and easily win the game.
This only works, however, if the first player has control of the center, or neither player has control of the center. If the second player gains control of the center, the first player in turn endures forced moves from the third move onward, until they stop with the second player having a free move; this move can always be used to immediately create a fork.
The dominance of the center in 3D tic-tac-toe, as opposed to it being merely useful in the traditional game, is because more immediate threats refer to it. The center square in 2D tic-tac-toe is part of four threats: each diagonal, and the two center-to-center lines. The center cube in the 3D game, however, is part of 13 distinct threats. Like in the traditional game, the center position plus any other position in the game creates an immediate threat; the extra move options in the 3D game, however, mean that a sequence of forced moves can always produce a fork position for whomever controls the center, while this can be blocked in the 2D version.
Winning the center-permitted game is therefore simple: take the center, then take any corner for which the block will not give your opponent a threat. The third move can create a fork, and that is that.
The center-blocked game has a different dynamic. There is now no single space that always produces a threat. However, two adjacent corners produce a special threat of their own. Take any corner; as a second move, take another corner such that there is one square that will block a connection directly between them, and it does not allow the opponent to produce a threat. (There would be no such square if the opponent could take the center.) The result is always a position that allows the creation of a fork, and in many cases allows two different forks to be created depending on the position of the opponent's first move. There are cases when even allowing the opponent to create a threat is acceptable, as the resulting block also produces a fork; those cases are optional, however, as there is no reason to permit this of your opponent.
3x3x3 3D tic-tac-toe is broken because whoever owns the center controls all of both players' moves. If nobody is permitted to control the center, then opening corner plays cannot be blocked as the first player can always steer around the second player's opening move. Neither of these strategies need to be very sophisticated, and the result is one of very few seemingly-playable games that are actually more trivial than the original tic-tac-toe.
The 4x4x4 game has none of these problems primarily because 2 pieces do not constitute an immediate threat; three pieces do, offering significantly more opportunity for interference, and more planning is therefore required to construct a fork.