I think it depends on what you're used to and how your body handles it. I'd take a rest day before an important race, sure, but I don't need them on a weekly basis. If enough of the training is easy riding, interspersed with racing/intervals/whatever, then for me its fine. If I wasn't doing the easy rides easy enough it might be different. And the running only days break it up. I seem to comfortably handle 2000 miles a month on the bike and running every other day.
I'm also mentally happiest training twice a day. If the pros can get away without any rest days, then I don't see the issue really. If I feel I need one, I take one, but not for the sake of it. Some people need one every week, we're all different. If I can see my HR is up I'd consider it, but it's currently 37 so not overly concerned in that respect. I know my own body pretty well and when it does need rest, I'm not averse to the idea, just not when I don't need it.
x post- I've spoken to a few elites/pros about this, and most of them don't have rest days. I was quite shocked that cyclists often don't have any sort of off season either, I thought they would, but loads just go straight from road to track to road etc with no break at all. I thought they'd take a couple of weeks at the end of the season.