A lot of Intel desktops these days have Intel AMT/ME. You can enable it and power up your system that way.
I have heard of that but can't seem to locate any free software for managing it. All I find are about 9 million different WoL programs none of which work.
On windows at the very least you'll need hardware that supports being put in a WoL state, a driver that supports the hardware being put in a WoL-enabled state, plus a BIOS that's capable of bringing up the system when it gets a signal from a WoL-enabled NIC. Usually none of these options will be enabled by default since they result in higher power consumption when the machine is turned off, and there's no unified interface for turning them on.
I seem to remember some bizarre problems with this under windows in the past (NetBIOS being one of them) and there were certainly a load of crappy NIC vendors whose drivers didn't support WoL under windows at all (and quite possible drivers for a NIC under Windows X might support WoL but the drivers for Windows X+1 might not).
WoL in linux is somewhat simpler in that the method for enabling WoL is universal and almost all of the NIC drivers support it, but you can still run in to issues with BIOS support.
This would all seem to indicate that, no, WoL is not OS-independent. I've already wasted enough time trying to make it work that I really don't want to bother with it; it seems to be way too finicky in the desktop market(and servers have BMC).
One thing I haven't seen mentioned and should work if you know when the machine is to be powered up next is to set an RTC wake alarm before you shut the computer down. IIRC if you set a "wake the computer to perform this task" you'll set an RTC wake in the BIOS and the machine will hopefully fire up and the correct time.
TLDR: WoL is dependent on hardware and software and depending on your env can be a PITA to get working. There's a reason why people prefer BMCs or baby-BMCs like AMT for remote power-on.
Unfortunately I don't have any kind of set schedule. In case you're curious, what I'm doing is I have a pair of gaming machines I use to multi-box on an MMO. I only get to use them sporadically a couple of times a week at very odd intervals. I used to just leave them on full-time, but that was eating up a lot of power; plus the room I run them in has no airconditioning and it's now summer.
Obviously its not a *huge* imposition to get up, walk into the other room, and hit a power button; but I'd really prefer to be able to turn them on and off remotely.
I'll start looking more into AMT.