I do it more or less by species, but with a bit of difference. I have three fly boxes that I carry. One has all my dries, soft hackles, nymphs, scuds, and such. It's split between the sides, with wets on one side and dries on the other. The second is all bass poppers, big streamers, and bass flies. And the third is all buggers in various sizes and colors.
All my boxes are organized in truly organized OCD fashion though.
I set mine to what type of fly like already mentioned, but on streamers, I have one box with RABBIT streamers, one box with SOFTHACKLES (like Gartside) and one box with WOOLLIES
etc. then the box with Chronomids, one for Small Dries, one for LARGE foam dries, Nymps, etc.
I then have one small box that I keep empty, and load it with what flies I think I will need for whatever water I am fishing.
I use both the species and the season when grabbing some flies for the pocket or boat bag. Even if fishing for King Salmon I'll always have a few dry flies and featherwings along in case I find more trout & char than salmon.
I haven't put this system fully into place yet, but I plan on setting up some boxes based on rod usage. I've been tying up some bigger and heavier flies that won't throw well on my 3 or 4wt that I normally fish. That said, it doesn't make sense to crowd those boxes with flies only castable on 6+ wt rods, or to frustrate myself by having the perfect fly looking at me with the wrong rod in my hand to throw it. So I think my new BuggerBarn will have quite a few larger floating flies as well, as they and the larger streamers will be attached to a 6wt.
I fish a lot of different water so I have a separate vest or bag for each type of fishing. Each 'system' has their own fly boxes, reels, and other necessary equipment needed for what I'll be doing
That way I can just grab and go.
For instance, in my trout tote bag, there's my loaded vest, boxes of extra flies, reels with different weight lines, my wading staff, extra sun glasses, reading glassed, extra tippet spools..etc.
And I have a similar set-up for salt, salt-kayak, warmwater-canoe, trout-canoe, and summer stream smallies.
By species: boxes for trout, another for bass and finally another for pike/gar. The atlantic salmon flies have their own fly wallet. I like to be able to take off on a moments notice, especially for trout.