I'm having the same problem and have been trying to solve it for over a week.
I'm connecting my Galaxy SIII to my computer's USB port via the USB cable that was included with the Galaxy in the factory package. I've tried this with both my work laptop running Windows 7 and my home laptop running Ubuntu (Natty stable).
When connected, Galaxy begins charging but there is no prompt nor anything in my pull-down notification bar to select MTP or PTP, and there doesn't appear to be a way to manually find the USB connection settings through the Settings menu (as I was able to do with my old Droid 2 if I accidentally dismissed the notification).
My computer does not detect that any device has been connected.
I have installed the correct Samsung drivers from the Samsung website on the Windows machine.
I downloaded and installed Kies software, which also does not recognize my phone. It continues to tell me to "please connect a device" and "when you connect a device, more features will become available."
I have tried shutting down phone, pulling battery, starting up again, and reconnecting.
I have tried opening Kies first and then connecting, I have tried connecting first and then opening Kies.
Nothing works. I just want to put some new music on my phone because I'm sick of listening to the music that was on my SD card when I switched it out of my old phone in August. I'm about to have to find my old Droid 2 and use it as a glorified SD card reader and just reinsert the card in that phone, transfer files to the old phone via computer, and then move the card back to the Galaxy. I'm a reasonably tech savvy user and this shouldn't be so difficult! I thought this phone was supposed to be the iPhone-killer.
ETA: I've also tried pulling the SD card and uninstalling a few apps like Advanced Task Killer and Titanium Backup in case any of them were causing a software conflict.
It seems like loads of people on the Internet have been having this problem since the phone first came out, but not everyone experiences it. I'm finding threads as far back as July and there's still no solution?