Hi johnny
You will probably enjoy Socrates Garden bar - it's off Socrates street in the Old Town ( where all the leather & fake shops are!) and is a really pleasant drinking experience, although like most in the main areas of the Old Town - not cheap! It's outdoors with lots of lovely plants & water features, excellent waiter service and the toilets aren't bad too ! Look out for the parrott (Socrates!) but he does nip you so don't get too close!
Yammas!!!

Either that or try the Mango bar http://www.mango.gr/internetbar.htm
on Pl Doreos next to the (nearly ruined) mosque - it's tucked away so good for a peaceful drink and is very reasonable by Rhodes Town standards (it'll be nowhere near as busy as it looks on the website!) - I'll certainly be popping in there next week some time