There really is no correct answer to that question. Obviously we don't all like the same weather, but as you know we go when we should get the weather we want and don't. The best we can do is look at the history of the weather where/when we are going and see what the average is, as well as what the records are.
I hate hot weather and don't mind some chilly or even fairly cold weather, but try to avoid real winter weather. Most of my trips I seem to wind up with lots of record heat.
That said since you asked about cross country eastbound from Oregon/Washington... The only trip I did that qualifies was the Trans America. We started on June 11, 2007. That was an okay date to start that year. We had no fresh snow, a few chilly nights, not much rain, only a little hail, and plenty of heat. We tried to schedule our start around when McKenzie Pass would be open to bikes, but closed to cars and were lucky enough to hit that mark.
On a different note... I went eastbound from San Diego from mid February and found that a good time for that route. Cool days and chilly nights with frost some nights and only a few really cold nights. Most if not all days got to over 50. I only remember one miserable riding day when there was pouring rain, a howling wind, and a chilly temberature that probably felt a lot colder than it was with the rain and wind. Mostly the riding weather was nice for someone like me who likes cool weather.
On another trip (in the Sierras), I headed out a time when I was as early as prudent for passes being open and was greeted with an early heat wave. We roasted on that trip.
