2001 triumph legend no spark in all three cylinders

Check, your spark plugs first, and if you have a spark at them. It can be a spark plug, plug cap, coil... I once had a similar problem with a broken spring in one of the cv carb diaphragm.
Franck

Check battery voltage, some 3 cyl. Triumphs miss on one when there is low battery voltage. T-birds for sure

Buy a wiring diagram and figure it out.

sounds like a timing problem, your manual will tell you how to make adjustments. good luck

If you can't get them from a Triumph dealer, try the web. Google motorcycle salvage triumph.

Check for water in the spark plug well. 
Probably worth pulling plugs and checking/replacing them. 

Long time for carbs to get gunked up if not put away with empty tank or special additives. Also clean splark plugs and load test battery. If battery more than 4 years old replace it.

this is most likely a bent  or broken shift fork  a broken gear cluster  or  foreign object lodged in the transmission housing  this is very complicated repair and should not be attempted buy untrained mechanic it can also be costly repair