My reply was on the contrarian side. Such advice, I think, has been taken too literally, and I find myself inserting that more often than deleting it.
From "Lapsing Into a Comma":
A misguided principle of the editing-by-rote school is to delete the word that whenever possible. It's often possible, but that doesn't mean it's desirable. Tin-eared editors chanting the mantra "Omit needless words" produce staccato ridiculousness that, in addition to sounding awful, can cause readers to stumble. Observe:He declared his love for her had died.
So you're reading along and you find that he declared his love for her. How sweet! Then you get to the end of the sentence and realize you've been misled. He declared that his love for her had died.
Believe is one of the big danger words for the that-averse. Often I'm reading about how the Democrats believe Bush (How sweet!), only to find that they actually believe that Bush did something wrong.
Or how about: They think Bush did something wrong. (Look, Ma, no that!) Are you shuddering at the word think? Don't be afraid. There's nothing wrong with think, just as there's nothing wrong with get. If your search-and-replace function is loaded to stick in believe and receive because the one-syllable words aren't good enough for edumacated individuals, well, cut it out.