In Hania we use Bank of Cyprus and have been very pleased with the service.
There's a list of branches (and ATMs) here:
There are pros and cons with all the banks. BoC has a good website with an English language version. In the branch we use there's rarely a queue and usually 3 - 5 teller positions working; you don't need to take a ticket as at National Bank or Tesco Deli counter; all the banking staff speak good English and so do most of the "back room" staff for things like mortgages, insurance etc.; they have Internet banking but only in Greek ("The English version will be ready in a few weeks" we were told at Christmas .... 2005 - still waiting so I've learned to do it in Greek, at least, to check balances and do inter-account transfers (fairly easy once you know that "logariasmos" is "account" as well as "bill").
Don't even thinking about getting a cheque book. They are available but very few shops will take them and people generally don't trust them. We have a savings account for which Debit/ATM cards were issued instantly (we were asked to wait for an hour before using them to be sure they were activated on the system) and a Sterling account into which our state pensions are paid (a "paper" transaction not an electronic one. The UK Pensions Service don't do direct electronic transfers to Greece) and an occasional cheque/money order (like Winter Heating Allowance).
I was given a Visa Credit Card as a "reward" for arranging a loan from the bank. The loan itself was agreed in principle within 30 minutes of walking into the bank intending to make an appointment to discuss the possibility.
Apart from that I've only ever had an account with National Bank - set up by our lawyer when we bought the house in '98 - which we closed immediately we came here permanently; waiting time in the queues could be a couple of hours. At least with the ticket system you could go shopping and come back.