Here is our (ACA's) stock answer on when to ride the GDMBR:
early summer to mid-fall. Caution: snow can occur at any time along the route. If it is a heavy snow year, high-elevation roads in the north may not be open until late June or early July
Sounds like a fun time to do it. Not saying you will hit snowy passes at all, but just be prepared to ether hike through snow or have some possible pavement re-routes. Especially as Joey said, in Colorado. It flurried on me in Northern Montana near Grave Creek around July 15th last year. Guess thats just the Rocky Mtns for ya