Can you provide more information?
Since you mention that is a "new" implementation, It could be a misconfiguration.
Brand/Model of those routers? The reason I ask for it, it's because for Cisco devices if you don't specify a VIEW it means is not restricted but in other brands such as DELL I found out that you must specify a view to make it works.
snmp-server group GRP_RO v3 priv access YOUR_ACL --> this would be enough for Cisco
snmp-server view READ_ALL 220.127.116.11 included
snmp-server group GRP_RO v3 priv access YOUR_ACL view READ_ALL --> this what I've got for the DELLs
Since snmpv2 works fine and it's all same protocol and port numbers, i'm sure it has to be your snmp settings.
Can you run the following command from your cacti server?It may differ from your settings so you may need to change -l, -a and -x for something different.
snmpwalk -v3 -l priv -u your_username -a MD5 -A your_auth_PASSPHRASE -x DES -X your_priv_PASSPHRASE 192.168.1.1 .18.104.22.168