From the Granada (for example), if you started walking south (against the traffic) on West Peachtree, then took a left at 14th Street (heading east), then took a right on Peachtree (heading south), it looks like the walking distance would be around 0.85 miles (according to MapQuest).
If this is further than anyone wishes to walk (or if it's pouring down rain, etc.), then my "interpretation" of the Marta online map is as follows:
It appears you would need to get to the Arts Center Station, and catch a bus for the Number 10 Route, which will head south on Peachtree. The Riviera Club says that they are "on Peachtree between 10th and 14th"; if my memory serves correct, I think the club is closest to the 10th Street intersection. On the map for the Number 10 Route, there appear to be stops at both 11th Street and 10th Street. Just tell the driver you're looking for the stop that's closest to The Riviera Club, and I'm sure they can help. I have tried (in vain) to verify this information with Marta; I've tried calling their customer service number (404-848-4711) multiple times, and there is no answer (nor is there a recording...it just rings and rings and rings). If anyone wants to look up the Number 10 Route for more information, here's the link to the 773 Grid on the Marta System Map:
Marta
If all else fails, call these guys:
Atlanta Checker Cab
Their number is 404-351-1111