* They are slow. In the beginning, they competed (effectively) against ships. "In July 1936, the airship also completed a record Atlantic double crossing in five days, 19 hours and 51 minutes." http://en.wikipedia.org/wiki/LZ_129_Hindenburg Today, they have to compete against jet aircraft - not effective.
* Given their immense volume and slow response times, they are very sensitive to wind taking off, landing, and when on the ground. This severely restricts their use under conditions that conventional aircraft have no problems handling.
Anyway, the reason we don't see airships much anymore is because for most things heavier than air craft are more economical. A Boeing 747 is a third the size of a Zeppelin but much faster, more reliable, and with much more usable space and lifting capacity. There's really no economic reason that an Airship should be preferred.
What about the fuel usage per unit of mass carried?
The US Military, on the other hand, has been increasing it's use of blimps for surveillance and data transmission/networking. These are useful in places were line of sight isn't available (mountains of Afghanistan, for instance) between radio/mobil cellular base stations. [1]
That and rising into the jet stream is a sorta abrupt introduction into 200knot winds. It would be disarming for most 'comercial passengers'.
[0] http://www.nature.com/scitable/blog/student-voices/the_rise_... [1] http://www.navysbir.com/n11_2/N112-169.htm
Fuel costs are but one consideration, and the same reasons we don't transport much cargo by sail are probably applicable.
They went bankrupt during development. I've heard that they couldn't find a solution for controlling the buoyancy, which is critical when you need to drop 100+ tons of cargo and don't want your airship to lift off into space. There's a video on Youtube in which they use water to control buoyancy of a smaller balloon [1].
[0] http://en.wikipedia.org/wiki/Cargolifter [1] http://www.youtube.com/watch?v=lralh-LwcJQ
It's only now that we have better weather info and new materials that anyone is taking them seriously again.