I am traveling to Greece, Italy, and other European Union countries in March. My passport expires in June. Do I need to renew my passport before I go? I'd prefer to renew it only when I need to.
Passport requirements vary from country to country, and whilst all countries in the Schengen Zone of the European Union have synchronized rules, these do not apply to all 'other European Union countries' as you've listed.
Specifically for Schengen countries, as of July last year there is a requirement for your passport to have at least 3 months expiry beyond the END of your travels within the Schengen zone. 3 months is defined as the same day that you will leave, 3 months later.
As you've only listed the months it's not possible to tell if you will have 3 months spare at the end, but at best you're cutting it very fine. If your passport expires June 15, then you will need to have plans to leave no later than the 15th of March.
Some other European countries require 6 months validity, whilst others such as the UK normally require zero.
Personally I would strongly suggest getting your passport renewed.