Does Jewish tradition or history give any clues as to what the breastplates look like mentioned in Exodus 28?

Victor L. Ludlow: Because the breastplate of judgment and its accompanying Urim and Thummim were sacred, later Jewish traditions emphasized their role as instruments of judgment and decision rather than their physical description. However, Maimonides, the great medieval Jewish Talmudist and philosopher, and some Talmudic sources, do provide some ideas as to the possible physical appearance of the breastplate.

According to these traditions, 12 stones, each engraved with the name of one of the 12 tribes of Israel, were the focal point of the breastplate. Each stone was square and set in gold, attached to a fabric skillfully woven out of 28 threads. One thread of pure gold was spun or folded with six threads of sky blue to make a larger thread. A second gold thread was joined with six purple threads, another with six threads of scarlet, and another with six threads of fine linen. Then these four larger threads were woven into a fabric one span wide and two spans long, a span being one hand width. The fabric, folded double, became a pouch, one span square. The four rolls of gold settings were attached to the outer surface, and the Urim and Thummim was placed in the fabric pouch.

A gold ring was placed in each corner. The two upper rings were attached by gold chains to similar gold rings on the shoulder of the ephod (shieldlike garment worn over the robe). The two lower rings of the breastplate were joined to gold rings of the ephod near the waist by blue ribbons.

When clothed with the breastplate and the other priestly garments, the high priest was prepared to perform his functions. The breastplate fulfilled one important role that Mormons would usually associate with the Urim and Thummim. When the king, the head of the Sanhedrin, or some other accepted person had a special question, he would go to the high priest. The high priest would then look down at a breastplate to see which of the engraved letters would shine out most brightly, and he would then construct the answer out of these letters. For example, when David asked the Urim and Thummim if Saul would continue his pursuit, the high priest Abiathar saw three bright letters: Yod in Judah’s name, Resh in Reuben’s name, and Dalet in Dan’s name. Thus the answer was given: YERED, “He will pursue.”

Jewish tradition indicates that the Urim and Thummim ceased to exist when the first temple was destroyed and the Jews were taken captive.

Centuries later, when the Jews were scattered across Europe, many Jewish communities made breastplates and placed them in front of the mantel of the Torah in their synagogues. These symbols, similar to the breastplate of the high priest, would often contain reproductions of the 12 precious stones. Otherwise, the breastplate and the Urim and Thummim are lost to the Jews.