I've tried a bunch of AdSense plugins but I can't seem to make any work. Can someone recommend something that is known to work?
My requirements are simple: display an ad under the header.
If your requirements end at "display an ad under the header" then just open up your theme's header.php file and paste your adsense code there directly and avoid using a plugin all together.
I use Advertising Manager on my site for AdSense and haven't had any problems with it. It has a useful import feature where you just paste your AdSense code and it will automatically configure the appropriate ad settings.
To insert an ad into the header using Advertising manager, you could simply edit your theme and use the following code:
Advertising Manager also offers a drop-down selector for inserting ads into posts and a sidebar widget.