If you don't want to use their, and the gender of the person is unknown, the options are limited. It's even more limited with the specific example sentence chosen than it might be with a different sentence.
The most common gender-neutral, singular, third-person possessive is one's.
However, the example sentence in the question can't make use of it directly:
✘ The black player's queen was attacked by one's opponent's rook.
This doesn't work because one's doesn't obviously correspond to the black player's. One's effectively means somebody's, but the sentence already establishes a specific person—the black player, so one would logically have to be referring to somebody different; however that doesn't make sense.
Note that a sentence that uses one's correctly is at the end of this answer. However, it also demonstrates that it can't effectively preserve the intended meaning of the example sentence.
We could use the following sentence:
？ The black player's queen was being attacked by the opponent's rook.
However, that has a bit of a semantic issue. The subject of the sentence is not the black player but the black player's queen. By mentioning the opponent's rook, it's assumed that it's mentioning the black player's opponent's rook. But, in terms of the actual syntax, it's literally referring to the black player's queen's opponent's rook.
This can be made clearer if the pronoun it is used:
？ The black player's queen was being attacked by its opponent's rook.
The most natural parsing of this is that it refers to the black player's queen, and not to the black player.
To avoid this problem with the subject, the sentence could be rephrased in a couple of different ways:
The black player's queen was being attacked by the opposing rook.
This version doesn't use a possessive in the final part at all, but it's fine because it doesn't need to. The white rook is opposing the black queen.
The black player had a queen that was being attacked by the black player's opponent's rook.
This sorts everything out, but it's awkward. It's the use of a pronoun that would avoid the awkwardness.
Note that if their were to be used, it would be used in this version of the sentence, not the one as written in the question itself—because of the problem with the subject in the original sentence:
The black player had a queen that was being attacked by their opponent's rook.
It's still theoretically possible to interpret their as referring to the queen rather than to the black player—but it's very unlikely.
A stylistically better version, that actually preserves parallelism, simply uses white player's:
The black player's queen was being attacked by the white player's rook.
Or we can use the generalized one with a rephrased sentence:
One should be careful that one's queen is not attacked by one's opponent.
But this is more than just a subtle change. Note that the sentence has moved from being about a specific situation to being about a general statement of strategy.
This is the problem with using one. Its use is generalized, rather than specific.
So far, in short, if you don't want to use their and you need to refer to a specific situation, there are really only two versions of the sentence that are natural—and neither of them uses a possessive pronoun (although one does use a possessive):
✔ The black player's queen was being attacked by the opposing rook.
✔ The black player's queen was being attacked by the white player's rook.
The only other option, one which I don't personally like but which many people have no problem with, is to use the following:
✔ The black player had a queen that was being attacked by his or her opponent's rook.
His or her has been a traditionally normal way of expressing an unknown gender while still making use of regular pronouns that don't cause some of the issues seen here. (It also has its detractors for several reasons: the use of a phrase rather than a single word, the ordering of the words, and the implication of a gender binary.)
Of course, this might all be a moot point in the real world, where the names of the players are actually known:
Jims's queen was attacked by Jane's rook.