What are some places that pay independent game developers for their games?
What's your game written in?
Coming from a cell phone game background there's sites like http://www.didmo.com/community/ that will host your games for you, they add an ad wrapper around it to make money, and you split the profit.
If you're not restricting yourself to a specific platform, there are a variety of content-delivery services like Steam (PC), PlayStation Network (PS3/PSP) and XNA Creators Club (PC/Xbox 360). These have various mechanisms in place to host independent games via their services.
If you are, then some of the other users here may have some good suggestions for other options for you :)
I'm guessing your looking at developing flash games seeing as you mentioned Kongregate, I looked into this a few months ago and the other site which looked promising was crazy monkey games. Hope this helps and good luck!
Kongregate is nice because they do revenue sharing and have contests with prize-money.
Don't forget that you can also port your game to the iPhone and put it up there, as well as porting it over to the Android platform and put it up on the Google storefront with little to no trouble.
You should look into XNA Creators Club though for the 360. Also, if you do it in XNA there's no licensing fees or anything, you can sale your program as is for windows machines. GoDaddy, a shopping cart with paypal integration isn't that hard to do, you could host it yourself.
Kongregate is very popular however; Their payment plan may not be what you're looking for and if your game doesn't become very popular then I don't think you'll be paid very well.
With that said there are other options you can consider. Have you looked at advertisement inside of your game? Many popular flash-based games have advertisements that display while the game is loading and/or for a fixed period of time. Every time someone looks at and/or clicks on and ad, you'll get a small amount of money they eventually will add up for some decent cash.
Unless your game is extremely popular (i.e. Desktop Defense), then I don't think you'll be able to make very much money off of it.
Have you considered hosting the games yourself, and merely placing ads on the pages?
I'm not trying to be flippant, I'm curious if you have gone down that path and what, if anything, you learned.