This is a very good question, that moves away from english language to embrace language and writing in general.
In Italian, my native language, we have adverbs with a very clear ending, "-mente" (i.e.: "lentamente = slowly"). Of course repeating many words with the same ending sounds dull, and should be avoided.
I have never read about a generic anti-adverb rule in italian. Consider that our language tradtion is traditionally much more rethorical than the english one: we use a lot of subordinates, implicit forms, adjective couplings, and therefore we are not short on adverbs.
After reading writing essays in english, where the usage of adverbs was discouraged, I realized that even when writing in italian I should have used them less.
But rather than just blindly expunge them from the text, you should think about what are adverbs. Grammatically, they are just specific words: the problem here is their function, not the word per se. If you write
He was walking slowly.
He was walking in a slow pace.
it's kind of the same thing, right?
There is nothing wrong in using adverbs, they are part of the speech as anything else. So why we suggest to avoid them?
Because they are at the same time cheap and redundant, and we can write a much better prose if we find a way around them.
Adverbs are cheap, because adverbs are generic, and can be very trite and devoid of meaning. What does "slowly" mean? Is it just a matter of speed, or is there is something more? Adverbs are cheap, because you use one word for something that could be shown with more nuances and details. Consider the "Show, don't tell" principle.
He was walking slowly.
He was walking with hesitant steps, as if he was afraid to hit the ground each time.
Adverbs are redundant, because most of the time don't add anything to a description.
He was running.
He was running fast.
Unless we have a very specific reason to indicate that "fast", we pretty much have everything we need in the verb "to run".
In conclusion: there is not a specific suggestion against adverbs in the italian language, other than to avoid the repetition created by the identical endings in "-mente". But to carefully use them in a prose that aims to be effective, is an advice that is always good taking in consideration.