Architecture communicates cultural values
This the correct version.
It's the difference between a direct object and indirect object.
For 'communicate', what you're communicating about is the direct object, so you should not use a preposition such as 'with'. Who you give that information to is the indirect object, so you should use a preposition such as 'with'. In your example, the cultural values are the subject of that communication, and those values are given to anyone who looks at that architecture. So it would look like this:
The architecture communicates cultural values with its viewers.
Either can stand on its own, for example:
The architecture communicates cultural values
The architecture communicates with its viewers
But which object is direct or indirect remains the same.