I eat fish 2-4 times a week, and turkey about 3 times a YEAR. Otherwise, no meat. I'm very healthy. I almost never get sick, and on the rare occasions when I do get sick, I get well quickly. [knocks on wood!]
I used to get sick more often when I ate more meat, but I'm not sure it's a cause/effect thing. I made other positive lifestyle changes at the same time I cut back on meat (more regular exercise, more avoiding potentially-harmful chemicals), so it's hard to know what made me healthier. I'm guessing "all of the above".