Don't even need a compass. The rising and setting sun would tell you east and west. Mid day in North America would be tougher as the sun is nearly overhead and it's time consuming to determine in which direction it's moving. Crossing the Mississippi would be hit and miss finding a bridge for bikes. Eventually you'd hit the Pacific, somewhere between San Diego and Seattle. Might be dangerous crossing sections of the SW desert in summer if you veer in that direction. Dead end roads would be tiresome. It would help to have a general knowledge of the geography and terrain features of the U.S.
I love and read maps so have a good handle on the layout and directions of the National Routes, I.E. US 20, US 30 etc...And would pick one and use that. I think 20 would be a workable route.
It would be an interesting challenge.