This is a point that has always confused me about open source software. Normally, I write everything from scratch. What I'm trying to find out is what licenses allows me to do this?
Why would you want to do more work? Of course you should. All you have to do is redistribute the OS technology source with your app (I am not a lawyer, but thats how I understand it).
The assumes
1) You are talking about an established open source solution, like hibernate, that you can reasonably assume works well.
2) The product you are developing is not using the open source technology as the 'secret sauce' that is going to make you money. Else you might have to open source that special part of your app.
Stay away from the GPL (LGPL is okay) and you won't have any problems. If you want to include GPLed packages in your application, things get tricky. BSD and MIT style licenses will get you the fewest obligations. In general, find the software package you want to use and read the license. They're usually pretty straightforward about what is and is not acceptable to do.