We have only two kids, but we're fortunate to be gifted a 2004 Sienna that I dearly love.
I agree that a lot of the medium sized cats get almost as bad of mileage. I drove my mom's CRV, a considerably smaller car, and was amazed that it got only 2 mpg better than our van.
Given that we only have two kids I have no doubt we will replace the van with a smaller car when it dies, but with three kids, I would stick with a van. Keeping in mind that your kids will be 15, 12, and 10 during the life of the next car, you need something that will hold three adult-sized people in the back. Maybe a larger sedan or wagon will serve for your family, it depends on how/where you drive. How many long road trips, camping, etc, or just short in town trips?