You need to get Triumph Shop Manual or have your dealer check the headstock bearing pre-load.
It's very important that the starting torque (pre-load) is set properly. There needs to be a certain amount of resistance in the steering head (headstock bearings). The easiest way to ascertain if, indeed, there is the proper pre-load is to lift the bike and get the front wheel off the ground (our famous Sears or Hardon Fright floor jack is the ticket). Put a small block off wood under the leading edge of the pad so the bike is slightly higher in the front than in the back. Get the bike off the ground and move the bars full lock right and left, then center the wheel. Apply a little pressure with your fingers to the RIGHT handlebar end, pushing on the bar to cause the front end to swing LEFT. If the the front end just flops to the left side after it is off center, without any apparent resistance, there isn't enough pre-load. Repeat on the other side was well to be sure.
You have to have the pre-load or starting torque in the headstock bearings. Without it, you will get what you are describing......headshake.
If you go by the shop manual for the R3 (I suspect the headstock bearing adjustment is universal for all models as my other bikes are basically, all the same too......
Slacken the locknut and adjuster nut....
Adjust the bearing free-play as follows:
a) Tighten the adjuster nut to 40 Nm
b) Slacken the adjuster nut fully
c) Re-tighten the adjuster nut to 6 Nm
d) Loosen the adjuster nut 90 degrees...(there will now be free-play)
e) Securely hold they adjuster nut in that position and tighten the locknut to 40 Nm.
Ensure that the adjuster nut does not move as the locknut is tightened.
Thats' right out of the Factory Manual, Section 14.16.
Setting the headstock bearings is just like pre-loading the bearings in any roller bearing application.