That was a problem for me in dieting in the past. Haven't had the problem this time. Two things made the difference, IMO. (1) Enough food to actually keep my digestive system going. I noticed in the other thread that you mentioned 600 calories today...I used to really restrict my food intake (albeit not that much), and doing so caused things to really slow down in terms of getting through my system. This time I cut the calories enough to lose, but realized that cutting more calories does not necessarily mean you lose faster (2) As Ann says, FIBER! The daily requirement is suggested at 25g per day. I started watching that number like a hawk, making sure to get above it every day, and it really does make a huge difference. Night and day, seriously.
I did those two things differently this time, and this is the ONLY time out of I don't know how many that I have been able to achieve my goal (and, more importantly to your question, not be constipated). Coincidence? Maybe, maybe not. But I think they both contributed.
Incidentally, I read that water does not help, even though conventional wisdom says it does. However, on the internet, you can find anything to support any position, so take that with a grain of salt.
And now that you don't have to be perfect, you can be good.