Could the Wright brothers fly across the US today?
It's possible, but it would be difficult under today's regulations.

Restrictions (regulations):


Avoid any controlled airspace that requires talking to ATC
For the sake of argument I'll pretend we have no radio on board, so no class B, C, or D airspace.
(You COULD call the controlling agency and coordinate with them, but let's just pretend we don't have a cell phone either.)
Avoid any airspace within a mode C veil
Because presumably we don't have a transponder on board we can't cross mode C veils.
Avoid any restricted/special use airspace
Duh.
Comply with any other restrictions noted on the chart
I did this in 10 minutes - I may have screwed up a little.
Fuel Reserves
This is the one that makes it tough.
With no wind 110 miles at 40MPH (the cruise speed) is ~2.75 hours endurance - I assume that's to empty tanks (which is how it's usually quoted for aircraft).
The FAA requires at least a 30 minute reserve per FAR 91.155, which means we can fly for 2.25 hours (90 statute miles, 78.2 nautical miles) before we need to land. All legs need to be under that. (It's possible a flight like this could get a waiver from the FAA, but let's assume nobody applied for one.)


Restrictions (Other):


Refueling
Where possible legs should end at airports that have fuel available.
(This isn't a hard "must" since presumably the Wright 1B can burn whatever kind of gasoline we can find once we're on the ground).
Airports
I'm excluding private / restricted airports from consideration as landing spots (again, we'll pretend our pilot has no cell phone and therefore can't coordinate landing access at these fields).


Simplifying Assumptions:


Simple fuel burn
Not having numbers for this aircraft I'm assuming fuel burn will average out such that the aircraft's endurance on all legs is about 2.75 hours (78.2 nautical miles with no wind). This simplifies the planning
There is no wind
(Or if there is, it's always a tailwind.)
There's an aviation GPS (or something like Foreflight) onboard
Frankly I wasn't going to spend an hour looking for landmarks.
You could probably manage the flight by dead reckoning if you had to though.
The Wright Model B can climb high enough to clear any obstacles/terrain
Again, not having numbers for the aircraft (service/absolute ceiling) I'm just making the assumption to simplify things.




So if we put all that together can we plan this flight?
What do you think I did with the first half of my lunch hour?


This is one possible route - there may of course be others

I believe I met all the restrictions, if I screwed up I'm sure someone will edit the flight plan (or come up with a different route).



A Note on Performance
As I mentioned in the assumptions I didn't have performance data available for the 1B when I laid out this route. Some folks helpfully pointed out in comments that there are flying replicas of the 1B.  These don't appear to be "exact reproduction" replicas, but I was able to find some performance data on them.
Given the maximum altitude in the performance figures (2,000 feet MSL) it would probably not be possible to fly the route plotted above with the reproduction equipment available (several legs are well above 2,000 feet MSL), and I would assume the original 1B had comparable performance.