I have been doing this routinely for a few years now for business, at least once a month. I used to have a 5 year visa on my US passport which expired and I did not bother renewing it. On one visit, the officer even mentioned that I had used my other passport, smiled and stamped my Israel passport without a second thought.
One caveat though, about 5 years ago I entered GIG on my Israel passport and the officer there detained me, told me that I need a work visa and sent me back to the US! If they ask you why you came, just say you are traveling or you have a business meeting.
When departing, show both passports to the ticket agent and then the Israel passport along with the entry slip copy to the officer and be on your way.