Hey, Rolf: Are you sure you could not develop the fitness for some mountain grades? Avoiding them would mean missing the best this country has to offer--really. Also, I'm not sure it's even possible to ride coast to coast without climbing a few mountains. The entire west coast is blocked by at least one mountain range at any given point: the Cascades up north; the Sierras in the middle/south; and other ranges to the south!
A trans-continental crossing will involve, for most people, two to three months of cycling. That means that you'll probably hit a wide range of conditions. I think that the ACA "Southern Tier" route started very early in the spring--like February--might provide some of the most temperate conditions though you'd almost certainly hit some sub-freezing conditions through New Mexico. It is possible, however, to find lower, warmer routes that run pretty close to the Mexico border in places. After April, the southern US starts to get very hot and humid--the conditions I hate the most.
For my crossing last year, I started in the NE (Maine) on Aug. 10th and finished in Ventura, California, on Nov. 15th. I hit every imaginable condition with the exception of snow: hot/humid, windy, sub-freezing, perfect clear autumn beauty. Crossing a continent is a full-on expedition, so I think the variety is to be expected. You'll find plenty of hard times, but these provide excellent pub stories when the riding is over. Gusto, mate, gusto! Cast off your lines and throw yourself into the journey!