Just a compass and a bike?

I would worry more about being cited for indecent exposure (*) and not having anything to put air in my tires if they went flat than I would about getting lost or how long it might take.

Now if I could bring clothes, money, bike gear and interact with people, I don't think it would be too tough to look at the signs. A combination of following routes with signs that say "East" or "West" and calibrating with place names I recognize could lead one reasonably well across the US. This notion that roads somehow dead-end becomes less true when thinking of the of the larger routes. The harder part would likely be getting *out* of largest urban areas.

(*) Or depending on time of year, just plain exposure.