Sweat can be an issue if it's too warm out, but otherwise I manage dry feet. It's just about layering and making sure there is no easy path for the water.
Thin socks (or thick merino in winter) under long Sealskinz socks make them more comfortable and less sweaty feeling.
Tights go over the top of the socks not tucked inside. Shoes or sandals with waterproof covers (proofed nylon normally rather than neoprene, as neoprene can be too warm) go over the top of the tights, then Goretex waterproof overtrousers go down over the top of the shoe covers to my ankles.
Water runs down my legs on on to the shoe covers, and off rather than down my socks. If it does creep up over the shoe covers (and it is hard for it to past the cuff of the goretex overtrousers), or creep in through the big hole in the bottom of them, or soak through them, it has a long way to go before it can start to get down to my feet.
My feet do sometimes feel clammy, but they are always dry when I de-kit rather than going all trench-foot on me.
Goretex boots on the bike were a disaster - I could never stop water running in to them and had swimming pools. If it's warm and summer, so too warm for the layering, then ideally I'd just have bare feet in sandals and not care about getting wet as I'll dry out quickly.