The concept of a military order, of which the Templars were one, arose out of the crusades. They were explicitly Catholic organisations dedicated to pursuing holy war. As the Orthodox churches never launched any crusades, they did not establish its own military orders.

There was at least one order that, though founded by a Catholic ruler, did have Orthodox members: the Order of the Dragon. It was a monarchical order, and so the members were not professed brethren like the Templars, they didn't take monastic vows. But the Order was still influenced by military orders. It was established by Sigismund of Luxembourg in 1408 with the express purpose of defending against non-Christian enemies and schismatics like the Orthodox churches (despite the presence of Orthodox members). In practice the Order was more intended to secure Sigismund's rule in Hungary by surrounding himself with barons whose loyalties he bought with the offices of the Order and gifts of lands. Members were also required to wear black every Friday or have five masses said for Christ's five wounds, so there were some more monastic aspects to it as well. Its membership later included Orthodox magnates like Vlad Dracul, father of Vlad Tepes, one of the historical bases for Dracula.

Unfortunately if you want to read more about this, almost of the scholarship specifically dedicated to this order is in German or Hungarian, but they do turn up in some more general works like D'Arcy Jonathan Dacre Boulton, The Knights of the Crown: the Monarchical Orders of Knighthood in Later Medieval Europe, 1325-1520 (Woodbridge, 1987), pp. 348-55.

The Hospitallers also had some Orthodox links. After the Hospitallers were expelled from Malta in 1798, the Russian tzar sheltered some of them and was soon appointed grandmaster. However, the recognised modern successor order, the Sovereign Military Order of Malta, is explicitly Catholic and chooses Catholic grandmasters.