When I start a new bar I will always encounter someone asking about "the other guy" who worked prior to me. I feel bad when anyone looses a job or whatever that case may be. I am always polite, NEVER bash anyone and simple respond with "they are no longer here and what can I play for you". I try and make the customer feel at home and convince them I will throw just as good or a better party then the guy before. When they start in with well "he used to play this or used to play that" I pay close attention on what the songs or artists might be and mesh that into selling them on how I run things.
Most times I can put them at ease and play something at least close to what they want and keep them there. Some DJ's send in friends to ask about where they are and try and bait me to say something bad or have that "you suck" look when they look at me. It does not bother me because I am there and the "other guy" is not