Hi there, are you using a desktop or a laptop?
Here is the thing,
I use a program called as atomic alarm clock on my laptop.
It has an option - "wake computer from hibernation"
So here is what I do :-
set the alarm when I want the computer to wake up, put my laptop in hibernation mode, and I make sure that the power cord is PLUGGED IN right from the start, even when my battery is completely charged.
So at the specific alarm time, my laptop just wakes up from hibernation.
Now please note, that this works for ME, as CWWOZNIAK rightly said, the BIOS should support the function.
Have also tested it with my friend's laptops, and it seems to work.
My current laptop is a DELL INSPIRON 14R, with Windows 7 Installed.