Cisco recommends soaking in Everclear (or rubbing alcohol if Everclear isn't available, 91%, make sure there's no acetone in it) for a few minutes, then blowing it out. He recommends that one never dry burn an atty, but others disagree. Also, he recommends a few drops of pg or vg if you're not going to use it right away, which keeps the coil from rusting...
That's basically what I do, and haven't had any issues at all. I'm more likely to lose an atty because it comes apart than dies or starts tasting foul. It's easy, and I don't worry about leaving them to soak overnight if that's convenient.
I've also used the sonic jewelry cleaner, but then follow that with an alcohol soak to get all the water out of it. The alcohol evaporates quickly, the water doesn't. I don't know that the jewelry cleaner actually helps... Might be good for a particularly dirty atty, but I try not to let mine get there. I clean with above procedure about every 8-10 mL. I keep four in rotation and this keeps me set for about once a week cleaning. They can go longer than that, but the taste starts to fall off after that.
(oh, and all this is relevant to various Cisco 306 flavors (bridged, bridgeless, Aero) and the 357)