To check where the flightpaths usually are just search for your flight on FR, the paths for LH723 (the OP) were like this:
http://www.flightradar24.com/data/flights/lh723/
The MUC PEK flight usually goes over northern russia, like most flights to the far east.
Southern China (HKG) and all the asian destinations usually take a track south of the black sea and then over Iran to Afpak and India. Depending on the weather some flights are routed a bit more northerly.
BTW, the countries you fly over make a lot of money, for example a LH A380 flying FRA-SIN spends 50000€ in flying rights each time. This involves some care, Afganistan for example kept their airspace open and secure even when the Taliban was running the show because it was a good source of income.