CRTs aren't especially bad for your eyes, looking at anything close for long periods is bad for your eyes, so that includes CRTs, LCDs, books etc. As for headaches, that's only if your run a CRT at 60Hz. 75 Hz and above, and definitely 85 Hz, you won't get headaches.
The main area CRTs are still unmatched is black depth. Anytime you see an LCD claim to have a contrast ratio over about 1500:1 it's a lie, a con, it's dynamic contrast and the screen is dimming its backlight when there's a lot of black on screen. The actual static contrast ratio measured on a single frame will be around 1000:1 at best. Even with such tricks LCD can't get close to the black levels a CRT can produce. The very best black depth I've seen on an LCD is still probably 4 times brighter than a CRT can produce.
Also viewing angles, I've owned 3 IPS panels now, and while they're infinitely superior to TN panels the viewing angles still do not compare to a CRT.
I've also yet to see an LCD that doesn't suffer pixel blur due to the pixel response time. Forget the headline pixel response time, again it's a con, even the quickest 1ms screens (which btw can't do any pixel transitions in 1ms, it's total BS) will be nowhere near that quick with some colour transitions and you'll get overdrive halos as well.
Ontop of all that you've a good chance of getting input lag, especially if it's a HDTV.
LCDs do have their benefits - they're very sharp, power consumption is low, they're very bright (debateable whether that's a good thing), they don't take up a lot of space etc - but CRTs have them beat in most aspects of picture quality.