I know there were similar threads, but none of them fit my situation. I have had Samsung Galaxy S3 for 2 weeks. After I rooted my phone using Kingo Android Root none of my apps could connect to the Internet except Browser. Only tried connecting my phone via Wifi. I tried to delete/edit hosts.txt, but it didn't work. My another attempt was to sync Google, but some of the checked things didn't want to sync. After I unroot my phone everything works fine.
@edit Did it with your AutoRoot method. All apps have access to the Internet except those, that are Google-related, for example: Google Play, Google Remote Desktop.
@edit2 Everything works fine if I use 3G. When I switch to WiFi it can't connect again.
@edit3 For some reason I've set proxy in my Wifi and after i deleted it everything works FINE. THANKS FOR HELP.