The answer seems clear.
Tarski (1936): "Semantics is a discipline which, speaking loosely, deals with certain relations
between expressions of a language and the objects (or `states of affairs') referred to by those expressions."
Davidson (1967): "There is no need to suppress, of course, the obvious connection between a definition of truth of the kind Tarski has shown how to construct, and the concept of meaning. It is this: the definition works by giving necessary and sufficient conditions for the truth of every sentence, and to give truth conditions is a way of giving the meaning of a sentence. To know the semantic concept of truth for a language is to know what it is for a sentence -- any sentence -- to be true, and this amounts, in one good sense we can give to the phrase, to understanding the language."
Montague (1968): "semantics...is primarily concerned with the notion of truth (in a model, or under an interpretation)."
Lewis (1970): "Semantics with no treatment of truth conditions is not semantics."
Heim and Kratzer (1998) Semantics in Generative Grammar: "A theory of meaning...pairs sentences with their truth-conditions."
Portner and Partee (2002) Formal semantics: the essential readings: "At the most basic level, a formal semantic analysis postulates a compositional, functional pairing between syntactically analyzed sentences of a language and their truth-conditional meaning."
Büring (2005) Binding Theory: "The task of the semanticist is to devise basic meanings for the words of the language and systematic ways of combining them so as to arrive at intuitively correct truth conditions for entire sentences."
Schlenker (forthcoming) `Semantics', The Linguistics Encyclopedia: "Minimally, a semantic theory should specify rules by which the truth conditions of complex sentences are computed on the basis of memorized properties of words or morphemes, together with a specification of the syntax (derivation tree) of the sentence at hand."
I find similar things said in Chierchia and McConnell-Ginet (2000) Meaning and grammar: An introduction to semantics and in Larson and Segal (1995) Knowledge of Meaning (the aim here is to specify our tacit knowledge of an internal compositional T-theory).
In Partee (2004) Compositionality in formal semantics, Barbara Partee describes her first introduction to Montague's work at a UCLA seminar in 1968 (attended also by David Lewis). She describes how at the time Montague's approach was different from anything she had seen and reports that she was especially excited by "...the revolutionary (to a linguist) idea that the core data were the truth conditions of sentences. Suddenly there was a non-subjective criterion of `observational adequecy' for semantics".
Any good quotes along these lines that I've missed? I'd especially like a better quote from Montague -- but he doesn't provide much actual prose to choose from.
Who disagrees with this answer? The Davidsonian project seems to get a lot of criticism (e.g. Soames) but its unclear to me what the fundamental difference is between the Montagovian and Davidsonian approaches. And the Montague inspired approach seems to be the dominate approach of the actual practitioners of semantics.
Another related set of good questions are these: What are truth conditions exactly? What are the things that have to meet a certain condition for truth to be achieved? Worlds? Centered worlds? Contexts? Points of evaluation? And do the truth conditions of sentences have to be the same as the propositions expressed by sentences? What is the relationship between truth conditions and the things we say?