Grammar Patrol Alert

Yesterday I closed a post with the sentence:

They serve whomever pays their bills.

In the shower this morning, I started to wonder about that.  Not that I think about blogging a lot while in the shower, but, you know, sometimes blogging is kind of addictive...

When I was in high school, I knew a lot about grammatical rules.  Now, however, I just know what "sounds right."  Since They serve him sounds right, I figure that They serve whomever... should be right.  But Him pays their bills sounds wrong; thus, Whomever pays their bills should be wrong. 

I'm sure someone out there can tell me which is correct: They serve whomever pays their bills; or, They serve whoever pays their bills. 

I suppose I should just can the whole sentence, and start over.

The FDA seems to owe allegiance to whatever persons or groups give money to the FDA.