If you're starting in April, I'd plan to start in the east. It may still be cold and wet, but the roads will be open (with the possible exception of a late season snowfall).
If you're doing a coast to coast ride from California, you can't count on the Sierra passes being open, so the Western Express is out. It's going to be darn hot by the time you finish the Southern Tier, with an April start. If you stick with an Adventure Cycling routing, you're left with a Southern Tier to Grand Canyon Connector to Western Express to TransAm. Possibly doable, depending on snowpack and seasonal variation in the weather at the higher altitudes of Utah and Colorado.
And while you're not really limited to the AC routes, similar comments would apply to similar routes -- you'll want a diagonal, or at least a partial diagonal route, to get past the mountains and avoid the worst of the heat and cold. (N.b. -- in 3 months, you're going to be hot and cold anyway, but you don't have to be miserable or endangered!)