Obviously the problem can be solved in many different ways, including trial and error.
Here is a solution that uses only logical reasoning and minimal computation. It is not an efficient method for finding the answer; but it can be a good exercise in logical reasoning.
(1) The three integers have to be negative, since the product is negative.
(So for the rest of the analysis we can think in terms of positive integers, since it is easier to describe.)
(2) By very rough estimation, the three numbers have to be less than 10.
(3) 5 can't be one of the integers, because the product would have units digit 0.
(4) That leaves only 6*7*8 or 7*8*9 as the possibilities.
(5) Without doing the complete multiplication, 6*7*8 has final digit 6, but 7*8*9 does not; so it has to be 6*7*8.
And then going back to the original problem, where we knew the integers were negative, the three integers are -6, -7, and -8.