The Book of Mormon is rich with chiasmus, which is a symmetric structure that looks something like ABCDCBA, fairly unique to Hebrew culture. That article outlines several instances of chiastic structure.
Additionally, there are more Hebraisms scattered throughout.
Repetition of possessive pronoun: Possessive pronoun prefixed to each list item, for example, 1 Nephi 2:4, 1 Nephi 2:11, 1 Nephi 3:22 (first quote), Mosiah 4:6 (second quote)...
22 And it came to pass that we went down to the land of our inheritance, and we did gather together our gold, and our silver, and our precious things.
In English, we would typically omit these extra pronouns.
6 I say unto you, if ye have come to a knowledge of the goodness of God, and his matchless power, and his wisdom, and his patience, and his long-suffering towards the children of men; and also, the atonement which has been prepared from the foundation of the world, that thereby salvation might come to him that should put his trust in the Lord, and should be diligent in keeping his commandments, and continue in the faith even unto the end of his life, I mean the life of the mortal body—
Repetition of preposition: Similar as above, but needlessly repeating the preposition or prepositional phrase in a list. For some examples, see 1 Nephi 19:11, 2 Nephi 5:15, Jarom 1:8, Alma 31:24, Alma 44:5 (quoted), Helaman 10:5...
5 And now, Zerahemnah, I command you, in the name of that all-powerful God, who has strengthened our arms that we have gained power over you, by our faith, by our religion, and by our rites of worship, and by our church, and by the sacred support which we owe to our wives and our children, by that liberty which binds us to our lands and our country; yea, and also by the maintenance of the sacred word of God, to which we owe all our happiness; and by all that is most dear unto us—
Cognate accusative: Verbs and related nouns are often built from the same root. You can see this in 1 Nephi 14:7 (first quote), 2 Nephi 3:18 (second quote), Enos 1:13, Alma 1:1, among many others.
7 For the time cometh, saith the Lamb of God, that I will work a great and a marvelous work among the children of men; a work which shall be everlasting, either on the one hand or on the other—either to the convincing of them unto peace and life eternal, or unto the deliverance of them to the hardness of their hearts and the blindness of their minds unto their being brought down into captivity, and also into destruction, both temporally and spiritually, according to the captivity of the devil, of which I have spoken.
Again, in English, this construct is unusual.
18 And the Lord said unto me also: I will raise up unto the fruit of thy loins; and I will make for him a spokesman. And I, behold, I will give unto him that he shall write the writing of the fruit of thy loins, unto the fruit of thy loins; and the spokesman of thy loins shall declare it.
Suffixed possessive pronoun: The pronoun is suffixed to the thing possessed. For instance 1 Nephi 22:14, 2 Nephi 9:25 (first quote), 2 Nephi 10:8, Moroni 8:20 (second quote).
25 Wherefore, he has given a law; and where there is no law given there is no punishment; and where there is no punishment there is no condemnation; and where there is no condemnation the mercies of the Holy One of Israel have claim upon them, because of the atonement; for they are delivered by the power of him.
In English, we would use a different word to rephrase it ("his power" or, for below, "his atonement").
20 And he that saith that little children need baptism denieth the mercies of Christ, and setteth at naught the atonement of him and the power of his redemption.
Construct states (basic/material/possession): In Hebrew, there's two nouns with an implied of, but English uses an adjective and a noun. The basic construct state is seen, for example, in 2 Nephi 25:2 (first quote), or Helaman 5:28 (second quote):
2 For I, Nephi, have not taught them many things concerning the manner of the Jews; for their works were works of darkness, and their doings were doings of abominations.
To satisfy this first construct state in English, we would say "dark works" or "their works were dark," and "their doings were abominable" (noun and adjective).
28 And it came to pass that they were overshadowed with a cloud of darkness, and an awful solemn fear came upon them.
We might say "dark cloud."
It's also common to see this with materials, for example 1 Nephi 3:3...
3 For behold, Laban hath the record of the Jews and also a genealogy of my forefathers, and they are engraven upon plates of brass.
We would say "brass plates." This appears in several verses including 1 Nephi 8:19, Mosiah 21:27, and Helaman 2:7, to name a few. Compare with Genesis 35:14 (KJV).
The third construct state, then, is possessive, where the word "of" is implied instead of the use of a possessive noun in English. Seen in 1 Nephi 7:4, 2 Nephi 5:14 (first quote), Words of Mormon 1:13, Mosiah 12:33 (second quote), and Helaman 7:10. Compare to Genesis 24:47 (KJV).
14 And I, Nephi, did take the sword of Laban, and after the manner of it did make many swords, ...
In English we'd say "Laban's sword."
33 But now Abinadi said unto them: I know if ye keep the commandments of God ye shall be saved; yea, if ye keep the commandments which the Lord delivered unto Moses in the mount of Sinai, saying:...
Compare this one to Biblical references, for example Leviticus 7:38 which uses both forms, "mount Sinai" and "wilderness of Sinai." In English we might say it as "Sinai's mount" or "Sinai's wilderness."
Remember: Mormons will be quick to remind anyone that these scholarly topics, though interesting, have little significance in the LDS Church, which is primarily focused on the spiritual teachings in The Book of Mormon, and that seekers of truth—which we on this site are not—should pursue spiritual, not scholarly, evidence.