If you speak even intermediate Japanese, chances are good that you can figure out the roots of at least three of the four seasons.

Haru is the easiest, related directly and obviously as it is to MJ hareru, "be[come] fine" (of weather, real or metaphorical). The CJ form was har.e. It seems a little odd in the MJ context to use a verb as a noun like that, without any nominalization or RY-ization or anything to soften the blow (e.g. hare or haru-no or whatever), but in fact in CJ you often see verbs used like that -- nominalization didn't have to be specifically marked with a particle. So it may well be that initially people were just walking around saying "Man, I can't wait until the becoming-fine" until eventually it got turned into a noun because it was so useful. (There's also the possibility that they come from the same root but have a sibling-sibling rather than parent-child relationship, of course.)

(I should mention the possibility that it comes from haru in the sense of "swell, grow" (of plants), but frankly I don't think it is correct, and neither does OONO "editor of the Iwanami kogo jiten" Susumu, who mentions that accentual issues make this derivation unlikely.)

Before spring comes winter: fuyu. Seriously, you probably do know this one. It's related to hieru ("be[come] cold"). The CJ form was hiy.e. Which means it could also have worked like I hypothesized above: "Aw, crap, it feels like the becoming-cold already."

Before winter, autumn: aki. This is an interesting one, because there are two competing theories. The first is that it's related to the /ak/ of aka ("red"), akiraka ("clear"), akeru ("be[come] bright, dawn"). I don't know about the brightness -- seems like summer is a lot brighter than autumn to me -- but I suppose the autumn light does have a certain clear quality to it, and of course the leaves turn red...

The other theory relates it to CJ ak.i as in "be[come] full, satisfied" (which, amusingly, lives on in MJ primarily as akiru, "be[come] sick of"). Because of the crops and the last chance to fill up on snacks before winter, you see. I find this one a lot more convincing, as Japan has always been about eating a lot in autumn. Shokuyoku no aki ("wanting-to-eat autumn") is a set phrase with two orders of magnitude more google hits than "shokuyoku no [any other season]".

Then there's natsu, summer, which is both the hardest for a Japanologist to figure out and the most linguistically interesting. This page summarizes it pretty well: a lot of linguists see a relation between it and Korean nierym, Manchu niyengniyeri, and other Altaic-family words beginning with /n/ and meaning "young" or "fresh". The downside, of course, is that natsu doesn't sound anything like either of those words except for the /n/.

The other common theory links natsu to atsu-, a Japanese root meaning "hot", but in that case the issue of where the extra /n/ came from is a bit tricky. So is the issue of why there are no similar words with an extra /n/ at the front like that (at least to my knowledge.)

Personal note: the internet at my domicile is completely dead until Saturday.