I upgraded to 1.9.3.4 today using connect manager. Everything seems to be ok, except Cron stopped working. It doesn't error, or anything. I can't run it from the command line nor web browser with the cron settings disabled in .htaccess. Any ideas?
I figured it out... an old extension was breaking it.
The first rule, don't ever update Magento through connect manager, even worse on the production environment.
The reason why it's not working can probably be found in your error logs.
So, check following logs:
It must be in some of those. Let me know about your findings.