Are there mountains to cross?

Definitely Appalachian Mountains are on west side of NC and the  Great Smoky Mountains and Tennessee west of those.  Smokies have lots of twists and inclines. NC contains the highest mountains in the Eastern United States. ( Wikipedia)  so your goal is to stay in the flatter eastern area of NC.  A much flatter route to NC from CA would be to go south through Texas, Louisiana, Mississippi, Alabama, Georgia, South Carolina and then turn left to go north to NC.  The Ozark Mountains in southern Missouri and northern Arkansas make for some crazy inclines too--usually interstate 44 (which replaced old Route 66) has a third lane for large trucks and trailers to go up as slow as they want and then get out of their way going down!  If the south is too hot (this is summer right?) try a more northern route. From Albuquerque, head to Kansas City (Kansas)Kansas City (Missouri) and take I-70 to Washington, DC (though you will still encounter the Appalachians in West Virginia)  and then turn right and go south to NC.  if you decide you DO have to go through some mountains, consider a resource like this book on Mountain Directory for trucks and RVs.  Consider a road map/atlas with altitude, terrain,& interstate highway system marked on it so you can easily change routes.  You may already know the locations of the interstates, but if not, I hope this is helpful:  they are laid out something like a grid of intersecting lines or an Excel spreadsheet.  

• Interstate hiways going east-west end in "0" and the southern most, I-10, is along or near the Gulf Coast.  The next east-west interstate I-20 would be north of 10  and sort of parallel to 10.  I-70 is San Francisco to Wash, D.C. and  Interstate 90 or I-90 is Seattle to Boston.
• interstates going north-south end in "5" so highway 5 is near the West Coast from Seattle to San Diego; I-55 is Chicago to New Orleans and I-95 is nearest the East Coast from Maine to Miami.