"Well, Googling this, here's the first result: "Low Prices on Sword Of Laban Free 2-Day Shipping w/ Amazon Prime." Thank you, hyperactive search engine optimization bots. Undeterred, I did some good poking around the interwebs and especially the Maxwell Institute's archives. We know from the scripture that the handle was gold and the blade was steel, and it apparently was heavy enough to decapitate someone in a stroke. Beyond that there really isn't much to go on, scripturally or otherwise.
It's apparently rare to find swords since they tend to rust quickly, but an article by John A. Tvedtnes suggests the sword of Laban might have been similar to one archaeological find, the Vered Jericho sword. That sword is too rusty to show me much about its original design, (picture here) but it more or less straight and around 3 feet long. The earliest known curved scimitars from the Middle East date from only about the 9th century A.D., which also suggests that the sword of Laban was probably straight. And I think that's about all we know."
So it looks like all those church pictures are correct in depicting the sword as straight. Cool fact for the day!
*The sword of Laban refers to a story from the Book of Mormon. In 1 Nephi Chapter 4, Nephi is told by the Spirit of the Lord to kill Laban in order to obtain the brass plates, which contains the scriptures and the genealogy of the Jewish people, including Nephi's family. It is important that Nephi and his family take these records with them to the promised land, but Laban is a wicked man and has tried to kill Nephi and his brothers and stolen their gold that they used to try to buy the plates from him. Nephi does not want to kill Laban, but the Spirit tells him, "Slay him, for the Lord hath delivered him into thy hands; Behold the Lord slayeth the wicked to bring forth his righteous purposes. It is better that one man should perish than that a nation should dwindle and perish in unbelief."
Nephi uses Laban's own sword to kill him, and he takes it with him to the promised land. It is mentioned later in the scriptures, when Nephi says that he used the sword of Laban as a model to make other swords for his people and his brother Jacob mentions that the people love Nephi, in part because he has often wielded the sword of Laban in their defense. The sword is again mentioned as being passed down with the plates to later leaders and prophets. Also, when Joseph Smith found the golden plates that he later translated the Book of Mormon from, he also found, among other things, the sword of Laban in the stone box.