The "syntax is not semantics" principle in the Chinese Room Argument (CRA) is based on the relationship between the Searle-computer and the Chinese symbols. Searle correctly characterizes this as a formal symbol processing relationship wherein the Searle-computer manipulates the symbols purely syntactically, according their shapes alone, without doing any subjective interpretation of them. This formal relationship is the linchpin of the CRA and Searle's rebuttal of computationalism (aka, computational functionalism, Strong AI).
Turing machine (TM) theory explains why this "linchpin" is merely a special case, and it exposes the huge gap in reasoning that comes from ignoring the most important part of the picture: the program. For example, the theory highlights this telling discrepancy:
If computers lack internal semantics, then why must the
Searle-computer's programs be in English?
Searle never addresses this significant inconsistency in his position.
The Searle-computer is fully programmable, hence it is a universal TM (UTM). Every UTM has a two-part input: (1) a program and (2) a "nominal input" for the program to process. For example, if given program ADD, for addition, and nominal input "3, 4", the Searle-UTM would output "7". Because the digits "0-9" are just formal symbols to the Searle-UTM, they could be encoded as Chinese characters, and the Searle-UTM would still perform addition--just like the CRA. However, the same is not true for the Searle-UTM's other input, the ADD program. If it were written in Chinese, for example, then the Searle-UTM would fail.
Notice that the Searle-UTM can correctly process the Chinese symbols (#2) on a purely formal (syntactic) basis only because it also has a program input (#1) that is actually responsible for determining what to do with them. The program--not the Searle-UTM--determines how the Chinese symbols are actually processed, so the Searle-UTM need only manipulate them formally, acting as the program's vehicle or "middleman".
On the other hand, the Searle-UTM is the only thing responsible for correctly processing the program itself. The Searle-UTM must causally connect the program symbols with the physical entities and processes that they represent—a non-formal process that realizes symbolic representations as specific real events. Thus, the formal symbol processing that is Searle's linchpin, is just a consequence of the special relationship a UTM has to its nominal (formal) input, which is mediated by a program input that is processed non-formally by the UTM.
"It's the program, stupid!"
Q: What is a program?
A: It is the specification of how some TM works--a kind of blueprint for instantiating a TM that typically is non-universal.
Q: What happens when a UTM runs a program?
A: Two significant TM computations occur: (1) the universal computation instantiates (2) the computation of the program's TM. (The Searle-UTM can only introspect on the first one, his own universal computation, which entails reading the program instructions in English and executing them.)
Q: What, if anything, is happening semantically inside the Chinese Room?
A: We don't know because we don't know how the program works. It is useless to ask the Searle-UTM because he doesn't know either. He doesn't know if his program is doing a Chinese Turing Test or tic-tac-toe. He only knows about his own universal algorithm: "read the program and execute its steps on the nominal input". To know the nature of the computation responsible for the externally observed behavior, the only thing that matters is the program, and it is left unspecified.
Searle completely ignores or dismisses the second TM computation, which arises from the program. Nevertheless, its existence is a mathematical fact, not just some philosophical assertion. It does not depend on anyone's subjective opinions or intuitions. It only requires an objective understanding of how UTMs work. This also explains why the Systems/Virtual-Mind reply has been the most popular kind of CRA rebuttal: http://www.scholarpedia.org/article/Chinese_room_argument#The_systems_reply
Searle's obsessive focus on a quirk in the nature of UTMs is somewhat understandable because UTM-computers and programs are so iconic in our culture. In philosophical discussions, however, it is crucial to focus instead on TM theory itself and on general TMs, not just UTMs. The failure to do so means that the CRA fails miserably as a refutation of computationalism while spawning decades of fruitless debate in the process.
UTMs vs. General TMs
Q: Aren't UTMs "universal" (i.e., representative of all TMs)?
A: While a UTM can instantiate any other TM via its program, a UTM's own internal algorithm and specialized input for this universal programmability is highly specific and not at all representative of TM computation in general. Focusing on this as the CRA does is an unhealthy distraction.
Q: What can a general TM do that a UTM can't?
A: A UTM must always act as a rote machine. It must faithfully ensure that the same given program will function the same way every time. In general, a non-universal TM could change it's own behavior over time based on its input-output history.
For a more complete explanation and discussion of this topic, see the articles provided here: http://www.chineseroom.info/
EDIT: A later version of this explanation is here, several reply-levels down: https://www.reddit.com/r/askphilosophy/comments/50igj8/if_you_could_chat_with_john_searle/