To my understanding, Mormons are neither Catholic nor Orthodox. I believe that they are Christians, but I'm not sure if they are Protestant or not.
Are Mormons Protestants, or are they just "Latter Day Saints"?
No, not really. While you sometimes find them lumped together they are not a subset of the class.
In popular usage such as among modern secular journalism, the term "Protestant" has come to mean anything that doesn't properly fall under the umbrella of either Catholicism or Orthodoxy. In this sense where there is no "Option D, none of the above", it sort of makes sense to lump in divergent sects with the rest of Protestantism.
There is, however, a fundamental difference: their theological roots are not the same. The LDS church is what is sometimes referred to as a "restorationist movement" — a reference to their theology's claim to be restoring something that was lost. While they agree with Protestants that the Catholic church lost its way, their proposed solution varies radically from that of Protestantism.
The basic tenets of Protestantism as a movement include rejection of a few Catholic doctrines and ultimately the Catholic church's claim to apostolic authority. In rejecting some of the core tenets of Catholicism, it based itself instead on a core set of doctrines (often loosely summarized in the "5 solas") that all require a certain reliance on the Scriptures as authoritative and uncorrupted.
The LDS church on the other hand is fundamentally based on the premise that Apostolic authority was lost (much earlier than even the rise of the Catholic church) and needed to be miraculously restored. This and other revelations are believed to have been given independently (and in opposition to) the existing claims of the Bible and historical Christian churches (whether Catholic, Protestant or Orthodox).
In other words, while they would also protest against Rome, their basis for doing so is entirely independent of the Protestant reformation and theologically they should be considered a sect (or cult depending on whose definitions you're using) outside of the theological umbrellas of Catholic, Orthodox, or Protestant traditions.
Latter-Day Saints do not consider themselves to be Protestants. The most fundamental concept of the Protestant tradition is an attempt to replace Catholic traditions and Catholic theology with a reformed theology derived from reading the Bible and attempting to interpret it properly.
Latter-Day Saints believe that this is impossible to do successfully; that the Scriptures were given by revelation, and can't be understood in their fullness by the wisdom of men alone, but require the same spirit of revelation and prophecy by which they were given in the first place (2 Pet 1:20,21). Therefore, because the original Church had become corrupt and apostate, (Mormons agree with Protestants on this point,) what was necessary was not to reform the church and its doctrines, but to restore them in their purity, through revelation and direct divine intervention. The official LDS doctrine of the Restoration is explained on Mormon.org.
Mormons and (most) Protestants agree that this puts Latter-Day Saints outside the Protestant tradition.
We don't consider ourselves protestant, but the restored gospel of Jesus Christ, with all the principals, ordinances and priesthood power that the Savior established during his mortal ministry. 2 Thessalonians 2:3 refers to a falling away. The truth was lost and needed to be restored. Acts 3:21 speaks of a restitution of all things. We believe that through Joseph Smith, Jesus Christ restored his gospel in its fullness.
I'm a Mormon myself. I think the term is "Restorationist" as we believe that the fullness of the gospel had been lost from the earth, through a great apostasy. We believe that God restored the fullness of the Gospel to the earth through a Latter-day prophet, Joseph Smith, Jr., and that God continues to speak to His children through a living prophet today, the current one being Thomas S. Monson.
There is one dimension not mentioned which discriminates between Mormons and Protestants. This is the fact that the latter are Trinitarians, whereas the former are not, in its strict sense. This is the same reason why the Jehovah Witnesses are not considered to be Protestants either, not even by themselves.
Of course they are. Follow the history-
I understand that Mormon theology differs greatly from mainline Protestantism. But it cannot be denied that the PEOPLE who became the Mormons were Protestants first.
EDIT- Everything comes from somewhere. People are born into their parent's world. Smith's family, and the families of the other founding Mormons, were all raised in the Protestant tradition. Their own ideas and philosophies are baked into Mormonism. They didn't create a new religion, they created another branch of Christianity- one that grows from the Protestant limb, just as the family trees of the Founders grow from Protestant limbs.