What are specific requirements, a writer should follow, when writing something that is going to be (or just can possibly be) not only read but listened as well?
The best advice I can give: read aloud what you've written. When you write a book, you're writing for the eye. When you write an audio piece, you're writing for the ear. If it doesn't sound good when you read aloud, then it won't sound good to the reader.
Some specific tips
Be concise and keep sentence structure simple. We're listening, not reading so we've got to comprehend quick and fast.
Write the way you speak.
Write as though you're communicating with one person at a time, that's how people will listen after all.
Use more present participles than you would for a book or text-on-paper kind of piece. For a book, it's good to avoid the verb "to be" -- but for an audio piece, it works because we use "to be" in speech constantly.
You might want to first listen to a few audiobooks yourself, not just the great stuff like Jim Dale's Harry Potter, but head over to LibriVox and listen to some of those volunteer audiobooks, some are good, some less so. You'll learn a lot from other's mistakes.
Remember to let the narrator breathe. You can write really long sentences in a book, but real people need to breathe. Shorter sentences and judicious use of commas is handy. Narrators can figure out places to breathe but make their lives easier and do that work for them.
A simple way to hear how your story sounds is to let the computer do text-to-speach for you. It doesn't sound great but it will help you catch the big problems early.
If you have any unusual names, a pronunciation guide would be handy.
When listening, it's not a simple matter to flip back ten pages when you've forgotten who the heck Character X is again. One thing that may help is to remind the reader of a trait or profession or other identifying sign once in a while, especially if the character hasn't been seen much yet/lately. (Everything in moderation, of course — doing this too often can result in so-called "Burly Detective" Syndrome.)
I used to be big on very naturalistic dialogue, complete with ums and ers and half-finished sentences. After several years of listening to audiobooks, however, I'm starting to lean the other way. Having characters stammer and constantly interrupt one another can be tough for a reader to pull off well. So yes, write sentences that people might actually say, but also let them get their thoughts out, at least most of the time.
Similarly, snatches of dialogue, as some authors use to capture crowd scenes ("It's her!" "...hasn't been seen in seven months..." "...swore up and down she was dead!") or dialogue that's only half-audible to the viewpoint character, can sound silly even in the hands of a good narrator.
And it probably goes without saying that very visual devices like online chat or computer code, while they may provide some nice flavour in print, don't translate well to audio.
Addendum: Text that's only set off typographically is tricky too. There have been a couple of books where I've been confused by a paragraph-long flashback that was set in italics, or by a stream-of-consciousness passage without attributions. (The audio versions of several of Terry Pratchett's Discworld novels used reverb on the voice of Death, whose dialogue was always set in small caps, but this sort of thing is rare.)