This is getting interesting.
You're not incorrect when you state that gene flow connects all the breeds of dogs. My guess would be that this is one of the key reasons why nobody's bothered to try and split dogs into separate species. That and the fact that the various kennel clubs would be up in arms immediately. You're not entirely correct, however, either.
Let me try and clarify. If we look at the various breeds of dogs as distinct populations, and the generic "dog" as a metapopulation, it's pretty easy to tell that the extremes (great dane and chihuahua) are clearly connected by a chain of intermediate populations. If we lined up all these dog populations in order from smallest to largest, there is no question that adjacent populations would be able to freely interbreed (all other things being equal). Hence, there is gene flow from one end of the chain to the other. IOW great danes have chihuahua genes and vice versa, even though they likely couldn't mate directly. To borrow a creationist expression, "they're all still dogs".
However, and here's where it gets a bit sticky, when the degree of gene flow between two populations becomes "insignificant", they can justifiably be claimed to represent distinct species if the divergence between them is great enough. The trick, of course, is to determine when "significant" becomes "insignificant" (i.e., where to draw the line). As with so much of nature, the problem really defies crystal clear definition. If all our populations of the metapopulation "dog" were bent around in a huge circle where the great danes ended up standing next to chihuahuas, and given the rather obvious pre-zygotic barrier between them, any "splitter" taxonomist encountering them would be justified in claiming they were distinct species. Even if our taxonomist could trace the relationship through all our intermediate dog populations, s/he could still argue the case that the degree of gene flow between our extreme populations was "insignificant", and hence still claim distinct species. A purist "lumper", OTOH, would counter that unless there was a break in the chain (i.e., a disruption of the gene flow between the populations through extinction of an intermediate form or population or whatever), there would be no justification for proclaiming great danes and chihuahuas different species. A hypothetical paleontologist from the far future comparing the fossils of both would definitely conclude that s/he had related but "no doubt" distinct species (or possibly even genera).
Just to make things even more complicated, there are examples where interruption of gene flow ISN'T required for speciation.
Irwin, DE, 2002 "Phylogeographic breaks without geographic barriers to gene flow", Evolution, 56(12):2383—2394
quote:
The spatial distribution of genetic markers can be useful both in estimating patterns of gene flow and in reconstructing biogeographic history, particularly when gene genealogies can be estimated. Genealogies based on nonrecombining genetic units such as mitochondrial and chloroplast DNA often consist of geographically separated clades that come into contact in narrow regions. Such phylogeographic breaks are usually assumed to be the result of long-term barriers to gene flow. Here I show that deep phylogeographic breaks can form within a continuously distributed species even when there are no barriers to gene flow. The likelihood of observing phylogeographic breaks increases as the average individual dispersal distance and population size decrease. Those molecular markers that are most likely to show evidence of real geographic barriers are also most likely to show phylogeographic breaks that formed without any barrier to gene flow. These results might provide an explanation as to why some species, such as the greenish warblers (Phylloscopus trochiloides), have phylogeographic breaks in mitochondrial or chloroplast DNA that do not coincide with sudden changes in other traits.
Irwin has another article, on-line, that explains in great detail the issue of gene flow and ring species:
Ring species as bridges between microevolution and speciation.
Let me know if the above actually made any sense to anyone.