There is an awful lot of talk on BYC about "genetic diversity" and "inbreeding" amongst certain breeds.  This brings up an interesting point that begs the question, Is inbreeding harmful?  I think one has to first define inbreeding vs. line breeding.  Line breeding is what I practice.  My understanding of inbreeding is Brother-to-Sister matings.  Line Breeding is Father-to Daughter or Mother-to-Son (or GD or GS or GGD or GGS, etc.) From what I have read, inbreeding can be very useful to enhance certain traits.  However, it may also bring a fault or DQ to light that would otherwise possibly not pop up.
The reason that genetic issues come up with inbreeding (and actually would also come up with "line breeding" as you describe it) is that deleterious recessive traits, which might be masked in the parents, are more likely to be expressed or passed on without being expressed in the offspring.  So if you have a hen who is heterozygous for a recessive deleterious or undesirable trait x, that means she carries the gene but doesn't express it (Xx).  If you breed her with a roo who is homozygous for lacking that trait (XX), then you have 50% birds who do not carry that trait at all (XX), and 50% will carry it but will not express it (Xx).  Now in the second generation (F1) if you either a) breed the two offspring that ended up with Xx OR you breed the mother hen to her male offspring with Xx, you end up with 25% of offspring who don't carry the gene (XX), 50% who carry it and don't express it (Xx), and 25% who express the gene (xx) and hence either die, are malformed, or exhibit the undesirable trait.  In going 1 step further (F2 cross), if the trait in question is NOT deleterious (i.e. fatal), and the original mother hen OR the F1 parent expressing the trait were to breed with the F2 bird expressing the trait, you'd end up with 50% carriers (Xx) and 50% undesirable or malformed birds (xx).  
That is a very very basic description of the process - lots of other things, like sex-linked traits and antagonistic pleiotropy (a desirable gene is linked to a lethal or undesirable gene), complicate the picture considerably.  And I do apologize if I'm just repeating stuff you already know!  But I think the idea of genetic diversity is that if you introduce "foreign" genes to your gene pool, you are less likely to run into the problems of bad genes being carried and/or expressed - the introduced genes "dilute" the effects of the bad genes within the gene pool.  Of course, if the outside genes aren't also desirable traits, you end up diluting the good genes, too...
Another reason to emphasize genetic diversity comes up more in wild populations of animals, but I think it could be applicable to domestic animals as well - a diversity of genes allow populations to respond to environmental conditions like changes in weather patterns, diseases, changes in food resources, etc.  If a population is small and they all have very similar genes, the population is more likely to be unable to "respond" by having some individuals that can survive a sudden disease outbreak or a string of unusually cold winters.  Mind you, it's not the individuals "responding", it's just that when you have a group of similar animals with slightly different sets of genes, selection - whether natural or artificial - can work on them without wiping out the entire population.
Ooh, I forgot how much I love evolutionary genetics!!!  Sorry, I'll stop now.