The word "that" is an interesting word for writers. According to this thread on Yahoo Answers, you'll see what I mean. There's more here at Dr. Grammar. And this web site is interesting as well. Here's an example below.
In the following, does the word "that" belong?
"He said that we should go fishing."Answer:
This usage of "that" is correct and belongs.
"That" sets off a dependent clause but could be left off without ambiguity:
He said we should go fishing.So there you are. I could link to a dozen more sites that discuss this. I've read blog posts about it before and I've seen authors talk about it on social networks. Personally, I've always believed in word economy and I try to leave out any extra words I don't need while I'm writing fiction. One of the last things I do during my own editing process is to do a search for words like "that" and remove them from sentences where they are not needed. I'm not saying I'm correct. It's just my opinion.
But I've also had editors who replaced the word "that" after I'd removed it. Does it bother me? Not at all. And I don't think it would bother most readers.