It will still work, but with slightly reduced safety. If the hot side of the line somehow shorts to the computer case, the ground will cause the circuit breaker on your house to trip. Without the ground, the case could become a shock hazard under such conditions.
Rather than saw off the ground pin, I'd get one of those adapters that Richard J recommended. It's not any safer unless an actual ground is connected, but if you later want to sell the computer, it won't have a messed-up cord.
Another option is to get a GFI or AFCI outlet, and change out your 2-pin outlet. Even without a real ground, that will give you some shock protection. And you can plug in your other 3-pin things to the outlet, too, without hacksawing all the ground pins.