Without hitting on the SPINS for entering certain areas, (I've flown over/into places like Iraq, Afghanistan, Pakistan, Iran, China, and the surrounding areas in either military or civilian capacity), overflying these countries is not too unlike flying anywhere else.
For example, here's a flightplan I pulled up on a Delta flight from Atlanta to Kuwait, using flightaware.com:
UGAAA2 AHN J208 HPW J191 RBV JFK PUT
EBONY N81B YQX NATU ODLUM UN551 NIBOG
UN551 BESOP UP6 REMSI UL603 ARNEM UL620
TALEG UL620 RIGSA UL620 CND UN616 DINRO
UL601 ODERO UP975 EZS UG8 SRT UT37 KABAN
R784 NOLDO UP975 SIDAD
You'll notice the familiar Jet airways in the states, and then they're filed aross the North Atlantic on NAT Track Uniform, and then the rest of the way on domestic airway routing through Europe and the Middle East. The nomenclature of the airways may vary; for example UL620 is referred to as "Upper Lima 620". There's nothing cosmic about the routing through Iraq (KABAN through SIDAD), it's on standard airways published on Jeppesen charts anyone can obtain. You're generally in radar contact from the time you coast in to Western Europe until destination, although the sophistication of the ATC services declines as you head further east. You may find yourself making more position reports and giving estimates for future waypoints. Also, keep in mind that you may be flying in an area of the world where neighboring countries don't talk to each other, so you may be doing your own handoffs (best get in contact with the next country 10-15 minutes prior to their FIR boundry). There's somewhat of a language barrier, although I've found the English spoken by Arab controllers is very good and easy to understand (compared to former Soviet states like Kyrgyzstan--not to mention they use meters!). When you're flying over certain parts of China or Africa, you may need to use HF radio in order to get ahold of anyone in certain areas. Navigation is accomplished usually with GPS/INS backed up with traditional nav aids.
I hope that helps. A more basic answer to your question would be that we generally obtain overflight permits / diplomatic clearances to operate in the countries we operate in/over and comply with their respective airway structure. In the event we're engaged in an active air campaign against an adversary (and hence wouldn't be granted overflight permits), Hacker's answer would be more applicable.