I can confirm this on my box with a Danish keymap. Accented keys work in regular forms and URL address line but not in Docs.

When the accent is entered followed by the letter neither are added to the document. Any following unaccented letters are added normally. Seems like none of the accent characters can be entered into the document even as separate characters.

Chrome version: 53.0.2785.92  Channel: stable
OS Version: Ubuntu 16.04.1

A new version of Chrome (53.0.2785.116 (64-bit)) is out but the problem remains. Still the dead key accent keys don't work inside Google Docs (at least) :-(

I had this problem too. Google Chrome for Ubuntu x64 16.04.1, all dead keys stopped working for Portuguese-BR, only inside Google Docs. Worked fine in the address bar. The keys stopped working in the middle of one document, and I wrote several text files before, with no issues. 

When I press the dead key (`) for example, and press (a), the output should be (à), but no character appears.

After posting the thread on Google Forum, I received an e-mail:

"Google Drive isn't currently available for the Ubuntu. However, I'd recommend leaving feedback about it through the following steps:
Open Google Drive.
Click on the Menu.
Select Help & feedback.
Select Send feedback.
Click on Send."

With Google Chrome 54.0.2840.34 beta (64-bit) the problem is solved :-)

Dead keys for accents work normally within Google Docs.

Thanks, developers!

#5 The issue is not in Chrome, but in Docs.  There's a specific check for Chrome 53 on Linux that does something to IME, so you will not see the issue on 54 or 52.  I'll bug the Docs team about this, but for now, can you verify that the issue still repros on 53?

Thank you for your comments.
I said it was Chrome because the problem didn't occur in Firefox and also when I downgraded to Chrome 52 it went away.
Anyway, you're the expert and you're probably right because:
1. At work my Linux machine updated to Chrome 54 beta and was OK (that's why I posted comment #5).
2. At home I had Chrome 53 stable and tested as you now suggested and ... surprise ... there was no problem with the dead keys accents :-)
3. At home I have just now upgraded to Chrome 54 beta and everything is OK also.
I hope this helps.
Cheers

Looks like the change in Docs has finally rolled out, so marking this as fixed.  Thanks vdalves.vet@ for your feedback