Gman, I was having the exact same problems as you and I was tearing my hair out. But I finally got it to work following Seven_007's instructions exactly and IN ORDER. Plus, there are others things you should know...
First of all, under development I had Stay Awake checked. Make sure all three of those are unchecked.
Secondly, do the following in the following order. Plug cable into phone. Plug cable into computer. Go back to phone and drop down the notification bar and select USB connected. When it asks if you want to mount, click on mount. Go back to your computer and see if there is anything under the new "Removable Disk" you should now see. If yes, you're good to go. If not, one or both of the following may be your problem.
1) The cable you are using is the one from the charger that came with the phone. For me, this did not work. Also, if you try to buy a third-party cable from Amazon (which is a great place to get a cheap HDMI cable, BTW) apparently according to a lot of the reviews a lot of these cheap third-party cables also do not work. At the Samsung site they suggested you get a better transfer cable. I got mine from the Walmart online store for $14.88. Yeah, I know, not cheap. Not only that, but it is only a little over two feet long as opposed to the charger cable which is about 5 feet long. Bottom line though, is this Samsung cable worked. (It says U1 on the white sticker on the large USB plug as opposed to the U9 it says on the charger cable. I have no idea what this means except that you want one that says U1 I guess, or just get the one they have at Walmart which is made by Samsung.)
2) I tried every single USB port on my computer (I have a lot of them) but the only ones that would work were the ones in the back of the computer. Combine that with a cable that's only a little over two feet long and you have a pain in the rectum connection where I have to pull my computer out from it's resting place and place my phone on top of the computer or on the floor. But at least it finally works.
Hope this information helps.
BTW, if you are trying different USB ports, you can't just unplug one and try another one. Shut off your phone and reboot it. Then do the steps in the same order. The phone SD card must be mounted to the new USB computer port you try while it is connected to THAT USB port.
Also, according to the Samsung site...http://ars.samsung.com/customer/usa/jsp/faqs/faqs_view_us.jsp?SITE_ID=22&PG_ID=2&PROD_SUB_ID=557&PROD_ID=559&AT_ID=318820
"Samsung Android USB Drivers are not necessary for USB Mass Storage Mode on the handset."