I rode nearly the same thing as part of a longer trip this summer. I had a blast.
Ultimately, I think that area is best traveled by car and on foot in my opinion. The roads from vegas to st. george (assuming that is where you fly into) are less than ideal; you are basically on an interstate through the desert for 100+ miles, though there are side roads available. You may want to arrange transportation to st. george if you are on a schedule and time spent on this section ultimately takes away from time in the parks.
Once you arrive in zion, bryce, and GC, the roads ultimately lead you to magnificent views of those parks. However, to explore them further, you must go on foot.
I really did enjoy biking the roads to the grand canyon from the north, to bryce from the east (through escalante region), and around the grand canyon on the east side towards flagstaff (though that may not be part of your itinerary). However, I was more inspired to stay in each of the parks you mentioned and hike for several days then I was riding my bike between them. If you are considering this during tourist season, traffic may be an issue in certain areas.
As a side note: I enjoyed biking flagstaff -> sedona -> jerome -> prescott in AZ, though past that it quickly descends into what I remember as desert hell. Just in case you were looking to extend the ride a little.