Hi,

I am a US Resident Alien and have an apartment in Rome, Italy, which I rent out. I file an italian tax return and pay taxes on the rental income. Do I need to declare that income to the IRS on my 1040?

Yes, you need to file schedule E (Form 1040). Also file Form 1116 to claim Foreign Tax Credit.
Another filing requirement may be Form TD F 90-22.1. Your U.S. Tax Return: U.S. Citizen or Resident with Foreign Income (http://taxipay.blogspot.com/2008/03/us-citizen-or-resident-with-foreign.html)