That said,

A modest suggestion to improve early-twenty-first-century discourse: remove that said from the starts of sentences. The phrase is at least slightly pompous, signaling that the speaker or writer has said something and is now about to say something else, something — gosh — contradictory. And the phrase has at least a trace of Richard Nixon’s annunciatory “Let me say this about that.” That said now seems to be everywhere: watch just an hour of CNN or MSNBC if you doubt me.

Sometimes that said is unnecessary or nonsensical, as in this passage from a recent New York Times restaurant review by Stephanie Lyness:

The fish soup was good, too, particularly in combination with the croutons, topped with rouille and grated cheese, that accompany it. (That said, our bowl was delivered without the promised croutons and toppings, but once we requested them, our waitress returned with a newly warmed, fully garnished bowl of soup with alacrity.)

Deleting that said removes nothing of the second sentence’s meaning. If anything, the deletion improves the sentence by avoiding two contradictions: but our croutons and toppings were missing, but we asked for and got them.

In a sentence whose that said is not unnecessary or nonsensical, another word or phrase can more clearly signal the relationship of one statement to another. Consider these excerpts from a recent Times column by Paul Krugman:

Kudos to Mark Weisbrot for saying the unsayable, and making a case for Greek exit from the euro.

I agree with a lot of what he says, but am still not ready to counsel that step, for a couple of reasons. . . .

[The reasons follow.]

That said, Weisbrot is right in saying that the program for Greece is not working; it’s not even close to working.

It’s easy to devise different phrasing:

Still, Weisbrot is right in saying that the program for Greece is not working; it’s not even close to working.

Weisbrot is of course right in saying that the program for Greece is not working; it’s not even close to working.

Weisbrot though is right in saying that the program for Greece is not working; it’s not even close to working.

Is there anything wrong with saying or writing that said? No. But making explicit the relationship between two statements is a good way to make clear what one thinks. And when a phrase becomes overused and tiresome, avoiding it makes sense. See also simply put.