Give everyone a timeline with important dates marked on it. The beginning of a timeline is, say, "the date of conception". The end is "the date of birth". Mark other important times also, of course - the trimesters, the first neural activity, the first point where reactions are measurable, and so forth.
Have everyone mark down what they consider to be the last valid point of abortion.
Now you can generate a few interesting points - a point before which 99.99% of America is fine with abortions, and a point after which 99.99% of America thinks they're wrong. Or you can use other percentages if you want. Make it so that before-that-point it's legal, and after-that-other-point it's illegal, then require case-by-case checks for anybody between those points. Guaranteed to come up with a result in a small amount of time, of course.
This is obviously a ghastly idea because it's very democratic and we've already established democracy doesn't work. I can also think of half a dozen ways to screw up the voting without even trying hard. Not only that, but we all know the government is great about handling case-by-case things quickly.
Still it's an interesting thought. ;)