There are three ways to solve the problem:
1. Compose an abstract map from the text and search for a real-world match. This is how Mesoamericanists and other theorists approach the problem.
2. Consult latter-day revelation, realize that two key modern-day identifiers exist, and use those as placeholders when evaluating the text. This is how I recommend people approach the problem.
3. Pretend it doesn't matter.
1. An example of the Mesoamericanist approach is Brant Gardner's book, Traditions of the Fathers. On p. 129, he writes, "Locating the Sidon is important not only for understanding the geographical features of the Book of Mormon but because locating the Sidon also locates the land of Zarahemla." Resorting to their own abstract maps and searching for a real-world match has led Mesoamericanists (and many other theorists) to disagree even about the river Sidon, although many of them have narrowed it down to one of two possible rivers in Mesoamerica. But even if they could agree on the river Sidon, they have to adjust the text to make it fit their geography as Gardner demonstrates in his long discussion of "Directions in the Book of Mormon" starting on p. 129, and the Sorenson translation (headwaters of Sidon instead of head of Sidon, etc.).
2. Two sections of the Doctrine and Covenants use the names of Book of Mormon sites: Zarahemla and Cumorah, in Sections 125 and 128 respectively. Mesoamericanists claim neither is conclusive, but why is conclusiveness necessary (or even relevant)? Why not give them a try before rejecting them? They are canonized scripture, after all. (It's interesting that D&C 125 was never canonized by the Reorganized/Church of Christ and other restoration groups.)
It turns out that if you put Cumorah in New York and Zarahemla in Iowa, the geographical references in the text fit nicely. The archaeology, anthropology, and geology also match up with the seas, the narrow strip of wilderness, going up and down, the narrow neck, etc..
In my view, this is yet another demonstration of the superiority of revelation over academic guesswork, but that's just me. The scriptures give us the minimum essential information we need--information we could never get on our own--and then we use our own efforts to supply the rest. IOW, instead of relying solely on study the way the Mesoamericanists advocate, I propose anyone can get a better solution through this formula: seek learning even by study and also by faith, a phrase repeated three times in the D&C. (D&C 88:118, 109:7, 109:14)
3. Some believers say they don't care about Book of Mormon geography. Everyone has different interests, of course. Some people have a spiritual witness that supersedes any earthly knowledge. (That approach also explains why there are 1.4 billion Muslims and over 1 billion Catholics in the world.) If one major purpose of the Book of Mormon is "to the convincing of the Jew and Gentile that Jesus is the Christ, the Eternal God, manifesting himself unto all nations," how convincing is it when investigators (and members) are told "no one knows where the Book of Mormon took place?" Or worse, they are told it took place in Mesoamerica, which requires one to believe that Joseph Smith didn't know much about the Book of Mormon and that not only he but two of the Three Witnesses were wrong (or merely speculating, confused, etc.).
When I hear believers say they don't care about the setting of the Book of Mormon, I suspect that in many cases that reflects not so much apathy or a lack of curiosity but an understandable desire to avoid the geography debates. I think this is a bigger issue than many realize. More Church members are inactive than active, and most investigators don't join the Church. A major reason for inactivity and not joining the Church is disbelief in the Book of Mormon, and a major reason for disbelief is the question of historicity (i.e., lack of archaeological and other real-world evidence). Investigators and members are told the Book of Mormon is a true history. When they ask where it took place, the answer that "we have no idea" is not exactly faith promoting.
What makes that answer all the more unsatisfactory is that if we use the D&C in conjunction with the text of the Book of Mormon (and the unambiguous and consistent statements of Joseph Smith, Oliver Cowdery, and David Whitmer), we have a real-world setting that reaffirms faith.