Sorry I haven't answered sooner Des, had a hectic weekend. Superb question, by the way!
For those who don't know, the Knights Templar were Europe's answer to the sohei of Japan or the Shaolin monks of China. Their full title being The Poor Fellow Soldiers of Christ and the Temple of Solomon, they were founded at the end of the First Crusade by a knight by the name of Hugues de Payens. It was his vision to create a monastic order dedicated to the protection of Christian pilgrims in the Holy Land. However, for the first nine years of its existence, the Templars only consisted of de Payens and the eight knights who joined him originally and the order was on the brink of being dissolved for lack of membership. However, with the help of Bernard of Clairvaux, the Templars eventually began to flourish to become one of the most feared and respected fighting forces in the mediaeval era, spanning from Jerusalem to Scotland, and amassing unimaginable wealth and power in the process. In fact, we owe the modern system of banking to the Templars' invention. As Des rightly said, they were annexed to the Church rather than a single state, although they were at the most powerful in France. However, they eventually became too powerful and in 1307, key French knights in the order were arrested, tried and executed for heresy by King Phillip IV of France, bringing about open season on all Templars, effectively bringing their legacy to an end.
While a fair amount of information is available concerning the Templars history and structure, their training methods were never well documented and, as a result, it's largely a matter of guesswork. The knights of the order probably would've had fairly eclectic training backgrounds since they were of noble birth and would have received training in the martial arts from a very young age in their respective households. Any training they did as a group would probably have been in drilling battle formations and the like rather than learning combative techniques, which they would already be familiar with.
However, the knight, while forming the backbone of the Templars, was not its only warrior. The ranks were also made up of "sergeant brothers" and "turcopoles", both of who would've been of ignoble birth and may well have received training in combat within the order itself.
A sergeant brother was a mounted warrior, much like the knight, the only difference being the benefits he received from the order and his social status. He would probably have received training in the use of the broadsword as well as the all-important lance.
The turcopole was a local soldier, probably little more than a mercenary, who would most likely have been drilled in the techniques of the bill and, of course, the sword.
The actual techniques these men would've learnt is not known for sure, but some external evidence exists to give us a clue. As far as sword training, the manuscript known as the Tower Fechtbuch, or simply I.33, gives us some insight. While not a specifically Templar document, this is the earliest known European sword manual in existence, dating to about 1280, and shows techniques with sword and buckler that could well have been the kind of thing taught within Templar ranks. An electronic copy of I.33 can be found at
http://freywild.ch/i33/i33en.html
To my knowledge, there are no styles of martial art today that claim a link to Templar techniques. That said, it's possible that those groups currently resurrecting the European schools of mediaeval and Rennaissance combat are unwittingly learning techniques that were practiced or descended from those practiced by the Templars, particularly those studying English and German manuals where the use of the broadsword enjoyed continued popularity, despite the fashion for the rapier in other parts of Europe.