I had a similar issue with an iPhone 3G. In my case, the screen when activated was really, really dark (easy to mistake for off). Apple replaced it even though it was out of warranty.
If the issue persists, try to reset the iPod touch while it is connected to power. To reset, press and hold the Sleep/Wake button and the Home button at the same time for at least ten seconds, until either the Apple logo or battery symbol appears. If the battery symbol appears, continue to charge the device until it is fully charged.
Hello NeilN-thank you so much. After several try and going back and forth to apple (it's out of warranty and they don't wanna replace it, sadly), one day it now turns on. Now what I really want is to backup everything from my device to another computer, in case something bad really happens. Do you think this is possible?
I have found dropbox to be an invaluable tool to offload from the touch and iPad pictures and files. The app is good for ios 3.1 and higher. Depending on the level of your ios, iTunes may limit what you can back up. But with dropbox on your mobile devices and computers, its the best thing going. No wonder Google is planning its own version.