Like most things the secret is to plan ahead. Think what you are going to say and what point you want to convey. Remember your audience are adults ( assuming they are) and unless it may be advantageous for some reason dont indulge in too many technical terms that may be either misunderstood or misinterpreted. In my experience communicating with your passengers and crew and involving them in the process avoids anger building up in most people.
Most "bad news" results from delays of one sort or another. This may be technical or ATC related or problems with loading or any combination thereof. If the problem is technical it is important to get the message across that safety is paramount ( people accept this ) and how long the engineers believe they will need to rectify the defect, inform the pax that these times are only estimates and that you will ensure theyare kept informed as to progress. It shows they are not forgotten and that someone in authority cares.
With ATC delays inform them of the cause, explain how you and the operations department are trying alternative routes in order to try and achieve an improvement. Tell them when you will next give them a progress report even if nothing has improved. It then looks as if you are in charge and are activley trying to do something. At the same time you are keeping everybody in the loop even if nothing is happening in reality. Your passengers are going to be frustrated with any situation beyond their control, however they are unlikely to become angry if they perceive you and your crew are doing everything possible to achieve a result and that you do care about their comfort and welfare.
Although the words "unfortunetaly" or " I'm afraid" etc may have a certain negative association the use of these words need not unduly concern the speaker if they are used with care and in the right context. This is particularly so when they are subsequently balanced by a positive message.
At the end of the day you are not a policeman telling a distraught relative that someone they love has died. You are a pilot telling a large group of people they are going to be late getting to their destination. It only takes a bit of liason with the rest of your crew and the ground staff to make sure the passengers are as comfortable as possible. Keeping the passengers informed by sounding calm and in charge and supportive should then normally result in very little anger and even a surprising amount of understanding from them.
it's not rocket science only common sense.