Many (perhaps most) Chess sets come with a King piece that has a cross on the top of it. Is there any problem with owning or using such a chess set? Should one break off the cross?
There is no prohibition against owning idolatrous figures. However, there is a prohibition against gazing as such figures, which would in general prevent one from owning them. According to Shulchan Aruch (YD 141:1) it is permitted to gaze at an idolatrous figure that is not intended for the purpose of worship. (See Rama there who includes the cross as an idolatrous figure, but maintains that in circumstances in which worship is not intended, such as a pendant on a necklace, it would be permitted.) For more information, see here.
R' Aviner brings several sources to indicate that although it is permitted to own the cross, it is nonetheless proper to remove the cross from the chess piece. The sources are
Shulchan Aruch, Yoreh Deah 141:1, Chochmat Adam 85:1, Shut Zera Emet 2:45, Shut She'eilat Shlomo 1:326
See here for details.
See here an answer to that and similar questions by Rabbi Kaganoff.
According to the reasons we have applied so far, Zev may be able to keep his fancy carved chess set. No one worships the cross on the king, and one could, perhaps, argue that this is familiar enough that no one is led astray by these pieces. As mentioned above, it is meritorious not to have any images at all, and certainly not to have anything that is reminiscent of idolatry. Thus, there is good reason for the custom to break off the cross of such chess pieces.
On a personal note, my father (a rav), would indeed break the cross off the king on our chess sets. However, I never asked if he considered it a Halachic issue or just a "minhag tov".