Monogamous couple: Stop using condoms?

If my partner and I have been monogamous for three years (and I was a virgin before we met) and my partner tested negative for the HIV virus (twice, about two years ago, with a six-month gap between the tests) is it safe for us to use a birth control method other than a condom? That is, are there still any STDs that my partner might have without knowing it, that we should be worried about, even though nothing whatsoever has seemed wrong with either of us for three years and my partner has had several regular annual physicals in that time?