I was told when I was a Mormon that the other sheep Jesus said he would bring were the Nephites.  The Book of Mormon records a supposed visit of Jesus to the Nephites, apparently about a year after His resurrection in Israel.
Two things to notice about John 10:16:  one, Jesus said He would bring, not visit, the other sheep.  (I read Biblical Greek, and there's a significant difference in those two words. The thought of Him bringing or drawing people to Him is echoed in John 12:32, where He said that after He was lifted up from the earth on the cross, followed by His resurrection, He would draw all men to Himself.)
Secondly, He made a point during His earthly ministry to acknowledge that, in a human body, He couldn't go everywhere in His few years of ministry.  He fulfilled all the Old Testament prophecies made specifically to the Jews by spending all His “quality time” with them. (He did make some minor exceptions to this, as with the Samaritan woman, but that's the kind of God He is, who meets people where He finds them.)
In another of those notable exceptions, in Matthew 15:21-28, a Canaanite woman came to Him asking for healing for her daughter.  He first told her, “I was only sent to the lost sheep” (–notice here the sheep imagery of John 10:16–) “of Israel.”  However, the woman is so persuasive that He commends her faith and heals her daughter.
So, to recap:  Jesus was sent, according to Old Testament prophecy, to the people of Israel, who had become “lost.”  Then, after He was resurrected, He used the power of the Gospel (the good news about His death, burial, and resurrection) to draw all people, including “sheep” who weren't Israelite by heritage, to Himself.
For more information, see The Mormon Mirage 3rd Edition:  A Former Member Looks at the Mormon Church Today (Zondervan, 2009). Also available as an audiobook and as an expanded-text E-book for Nook, Kindle and other reading devices.