I want to replace the grass in my lawn with some eco-grass (a blend of fescues, not the astroturf), but I'd like to avoid using Roundup to kill my existing grass. I was thinking of either tilling my entire yard with a rototiller or getting a sod cutter and then flipping the sod over (so I don't lose any topsoil).

Do not use the sod cutter unless you get rid of the turf, turning the sod over will only cause it to start decomposing and you will have to wait 6-8 weeks before you can re-seed.
Tilling is a better idea if you do not want to use any chemicals. Good luck with it.