I have been a customer of ICICI bank and as per my experience, I would say, they provide very good customer service.
For most of the purposes you won’t even need to visit any bank branch unless you plan to deposit cash into your account.
The internet banking and mobile banking service are one of the best among the Indian banks as they are more user friendly.
The bank staff(like I have my account in Aundh(PUNE), Laxmi market(Miraj) and Dhamani Branch(Sangli) are very courteous and the best thing is the bank is never crowded(I never seen any).
The only limitations I saw is, they are charging heavy bank interest for non maintenance of balance, cheque bounce etc then any other bank.
I am loyal customer of ICICI bank since 2012 and was having Salary Account and now saving A/c and Current Account.