What makes you think zeppelins can fly in that stuff?

Because zeppelins are AWESOME, while jet aircraft, which safely carry hundreds of millions of passengers a year at near-supersonic speeds, are not.

Hmm, all aircraft are pretty awesome, but nothing is as crazy mad awesome as Zeppelins. And hey, if even the motors on Zeppelins get clogged in this ashmageddon, we'll just install rowing monkeys.

Hmmm... rowing monkeys.

Forever. Anything a volcano can do to an airplane's engines it can do to a zeppelin's. If it keeps erupting, they'll just fly jets at lower altitudes and burn more fuel.

Hold on a sec, I'm sort of serious here. A zeppelin is lighter than air -- it can just turn its engines off and drift. Plus they hardly need to be jet engines. A heavier-than-air plane is constrained to go constantly at a very high speed to maintain lift, and its failure modes are severe. A zeppelin can go much slower, so much simpler engines will do. I expect the ash is much worse for jets than for propellers, for instance.

However, the point about lower altitudes is well taken, alas.

The ash is a fairly high altitudes. Jet planes suffer a severe penalty when flying a low altitudes. Propeller planes suffer no penalty. The solution is to bring back propeller planes. they had more bathrooms, anyway.