As bad a rep as adverbs have, they're actually pretty handy during a first draft. They allow you to jot down how a character feels or how they say something without losing your momentum. You can keep writing, and go back and revise later.
They're wonderfully helpful red flags during revisions that point out "here's where you have a great opportunity to flesh out what your character is doing." They're like your brain telling you about the emotional state of your character, and pointing out a place you might want to examine further.
I walked cautiously across the room to the back door.Here, cautiously is doing the explaining, telling that this person is nervous in some way. You could find another word for "walked cautiously" like tiptoed, or sneaked, or slipped, or whatever, but that only solves the lazy adverb problem. It doesn't do anything to capitalize on what your subconscious might be telling you. Instead, try looking deeper and showing someone being cautious in a way that helps characterize and further develop the scene.
I scanned the room, checking for tripwires, pressure plates, anything that looked like it might be a trap. Looked clear. I darted for the door.Is it longer than the first adverb sentence? Sure, but it's more interesting and tells you a lot more about what's going on. Which probably saves you words somewhere else. Especially since there's a decent chance the description in that scene might be a little flat. If you had a better sense of the character's emotional state, you probably wouldn't have used the adverb in the first place.
Look at your adverbs and what those sentences are describing overall, and then think about other ways to get that idea across. It's not always about replacing it with a stronger word, though that certainly is an option. Sometimes those adverbs are pinpointing an important aspect that would really make the section sing if you fleshed it out.
Look at where you use adverbs and identify what you're trying to do with them. They're telling the reader what's going on, but if what's in your head doesn't make it to the page, you can wind up with a reader/writer disconnect.
"That's just wrong," Bob said angrily.Here, the adverb is used to denote anger, but it's a lazy word because it makes the reader have to decide what Bob's anger looks like and how he acts when he's angry. And readers might get it wrong. One reader might think Bob screams and yells, another might think he gets real quiet and dangerous. But if you think Bob cracks jokes so he doesn't blow up, what you write for him won't connect right with the reader, because they'll have different ideas in their minds and read the words in that context.
I'd always thought of adverbs as placeholder words, but they can also play helpful role in editing. They're not the enemy, they're just your subconscious telling you to, "do more here."
More on adverbs here.