You probably already know this, but it can't hurt to check the basics: many triples on bikes seem to shift best to the granny when the chain is on one of the innermost sprockets in back. I usually do that shift when the chain's on the biggest sprocket or one of the two next largest ones.

Also, the difference between sluggish and reasonably quick shifts to the granny can be a matter of repositioning the front derailleur so that the rear of the cage is just a degree or two further in toward the frame.

And, of course, make sure to confirm that the vertical gap between the bottom of the cage and the top of the teeth on the big chainring is as it should be. About a nickel's thickness, if I remember correctly.