How do the Chitauri "ships" propel themselves?

I wasn't able to find confirmation either way, but my belief is that they fly via mechanical means. The leviathans are organic creatures in a armored harness/shell. That shell's back fins have electricity discharging between the fins. Their arm fins have armor that protects the creature's arm, which they use to hack through buildings. Both of these could contain some of the same anti-gravity technology and propulsion that the Chitauri's small flying platforms use.

I found some discussion of their design to back up what I'm suggesting. MTV interviews the Visual-effects Supervisor of the film, Jeff White, here:

These guys are actually organic beneath all of that armor. They have these worn gold plates and energy signatures that ripple down the side.

(emphasis mine) Since they don't have any weaponry besides the physical damage they inflict, what energy would be rippling? I put forth that this is anti-gravity and propulsion energies.

This next one further confirms that they're organic underneath the armor. io9 interviews Production Designer James Chinlund and Concept Artist Ryan Meinerding here:

Meanwhile, the aliens also bring with them the Leviathans — those weird, serpent-like creatures that you can glimpse trashing the city in some of the trailers. Whedon wanted "a creature that was used by the aliens as a transport but also as a fierce attacking animal," says Meinerding. The concept art team on the movie created a number of versions of these beasts — many of which were rejected for being too mechanical.