How different is flying through Free Route Airspace (FRA)?
Short answer: We do nothing differently in the cockpit when flying in a FRA. The route segments are filed as a long direct and simply flown like that.
As ususal conflict prevention/solving lies with the controller. Only short term conflict solving lies with the cockpit crew (think TCAS RA). In a FRA this is nothing different. In a &quot;regular&quot; airspace with airways and waypoints the problem is the same. Only are your conflict hotspots concentrated on waypoints where you have crossing or converging airways.  This presents similar problems to the controller.
The FRA I mostly fly through is in south-east Europe (Vienna, Ljubljana, Bratislava and Budapest FIRs). Often the Austrian controller clears us from our western entry point on a direct all the way to the eastern exit point of the adjacent Budapest FIR. So there is some trans-national coordination going on in the background.
From the Eurocontrol Free Route Airspace FAQs:

Have ANSPs developed strategies to mitigate the potentially more
difficult detection of conflicts that comes with FRA implementation?
[...] ATC systems across Europe [...] include conflict detection tools
[...] using aircraft trajectories calculated by the ground system which
are updated in real time as a flight progresses.  [...]  The tools help
controllers identify conflicts well in advance of them occurring [...].
At the same time, the implementation of Free Route Airspace generates a
much better spread of conflicts compared to the concentration of
conflicts generated by the fixed route network.

source: http://www.eurocontrol.int/articles/free-route-airspace, omissions by me
Route deviations from cockpit side (to avoid weather for example) are handled like everywhere else. Ask the controller for an avoidance heading and you're good to go. Possible arising conflicts are then solved by the controller. Same works for conflict mitigation from controller side. You might get a heading until clear of conflicting traffic and then a direct clearance again.
In my experience this works just fine in the daily operation.