Does anyone know any good alternatives for flash to do simple animations for web sites? Haven't had much luck with my googling.
Silverlight jumps out as a flash alternative.
The main thing that you need to remember is that your users need to be able to view it. If it's some obscure plug-in, no one is going to download or install it. Flash is the main event with Silverlight in a close second. You can do some cool stuff with HTML5 but not a lot of browsers support it even partially yet.
Responding to the comment on MarkM's answer: you can create Flash animations in lots of ways without using Adobe Flash. I like and own SWiSH Max 3 from SWiSHzone. It has most of the power of Flash, including scripting, for $99 US. They have even simpler and cheaper tools, depending on exactly what you need.
There are at least 10 other non-Adobe Flash animation creators, but I've only used Flash (including Flex) and SWiSH Max.
Anonymous raises a good point. How simple? Do you program? Do you want to work with time-lines? Do you just need a bouncing ball or a spinner? Do you just want simple special effects on text? The type of animation, even simple, varies tremendously and your level of understanding of different techniques and approaches will also influence how best to approach the problem.
JavaFX will soon have a designer, will be incredibly easy to do animation, and is built on Java. The Java plug-in may not be as pervasive as Flash but it's, in my opinion, better than Silverlight. But i agree with CarlF that you might want to look for different ways to create Flash animations. There are tons of them. Without knowing the answers to the questions above, it's hard to answer your question. It might just take more googling and trying out everything you come across.
In my opinion, until most browsers support