What was wrong about it?

I don't think it's WRONG, but I would use BY instead of WITH and change the last part from a noun to a verb. "My dog woke me up by barking annoyingly".

andreajgb"my dog woke me up with an annoying barking"

I think the suspect is " with an annoying barking". Barking is non-countable noun, so "a" incorret.

Woke up with a headache -ok

With your sentence, I think "by" is a better choice. I would reword it as follows: The annoying barking of my dog woke me up. Or I was awakened by the annoy

I don't think you will ever hear a native speaker say "my dog woke me up with an annoying barking" it just dosen't track. Instead you might try "with his/her annoying barking."

And I don't think I would ever think to include the adjective or adverb at all:

My dog woke me up with his barking.-- This pretty much says it all.

So it isn't wrong, but you just wouldn't say it?

It is wrong, check Dimsum's reply.