First, you really should read the paper. Putnam is a scholar I have a great deal of respect for (I've read Bowling Alone twice) and he combines painstaking research with a genuine gift for communication that makes his arguments a pleasure to read, and renders any attempt I might make to summarize his findings both redundant and misleading. This is a complex, subtle piece of work that demands careful reading, not least because of the potential seriousness of its implications, and you owe it to yourself to really understand what Putnam is saying, instead of relying on whatever cock-eyed interpretation the popular press chooses to come up with.
That said, I have to say that I have some concerns with the validity of what Putnam is finding, or at least, with the validity of the implications he draws from his findings. When I first heard about the study, the first question that popped into my head was - what about sample selection? Obviously people aren't randomly assigned to the neighborhoods they live in, and Putnam is observing a cross-section, so he doesn't know how attitudes evolve as the composition of the neighborhood changes. Isn't it possible then, that the lower civic engagement he finds is a function of the kind of people who choose to live in more diverse neighborhoods, rather than of the neighborhoods themselves?
Not surprisingly, Putnam addresses this concern in his paper, but dismisses it more or less out of hand. He writes:
It's a strong argument, except that it seems to me that in making this argument, and in the study more generally, Putnam is mixing two very different attitudes. An attitude of distrust and an attitude of disinterest. The question that Putnam's runs his regressions on reads: "How much can you trust people in your neighborhood?". What would a low-scoring response on this question mean? It could mean that you actively distrust your neighbors, that you are (to use Putnam's phrase) "irascible and misanthropic". Or it could mean simply that you have no interest in your neighbors, that you are too caught up in your own circle of (non-neighborhood based) social activities, hobbies, interests or personal amusements to have bothered to meet your neighbors or even get to know their names. You're not paranoid or even necessarily an introvert, you're just not interested in your neighbors. Is it really that far-fetched? How many people do you know who live in Manhattan and have ever bothered to meet their next-door neighbors? Does this mean that everyone living in Manhattan is a paranoid introvert? I know I would have responded to that question by saying I trusted my neighbors very little, but it's not because I mistrust them (on the contrary, I'm sure they're really nice people) it's just because they're complete strangers.
People mostly choose where to live, and that simple fact opens up a hornets’ nest of methodological problems with correlational analysis since people with a certain characteristic may choose to live in distinctive areas. For example, the fact that people with children live nearer to schools does not mean that proximity to a school caused them to become parents. In our case, however, selection bias is prima facie implausible as an explanation for our results. For selection bias to produce a negative correlation between diversity and sociability, paranoid, television-watching introverts would have to choose disproportionately to live in mixed neighbourhoods. Phrased differently, a self-selection interpretation of our results would require, for example, that when non-whites move into a previously all-white neighbourhood, the first whites to flee (or the most reluctant to move in) would be the most trusting, and the last to flee would be the least trusting; or alternatively, that ethnic minorities and immigrants would selectively choose to move into neighbourhoods in which the majority residents are most irascible and misanthropic. Common sense suggests that the opposite is more likely; if anything, selection bias probably artificially mutes the underlying causal pattern. In short, taking self-selection into account, our findings may underestimate the real effect of diversity on social withdrawal
Now, imagine, for a moment, a world made up of two kinds of people - those who really value interaction with their neighbors and take an active interest in them, and those who don't. Imagine further that homophily holds and people are more trusting of / more comfortable interacting with people of their own ethnic group. Now, when the first people from some other ethnic group start to move into a previously homogeneous neighborhood, who will be the first people to leave? The people who like to spend time with their neighbors, of course - those who've never been interested in their neighbors will probably take longer to notice that a shift in demographics is happening and in any case are unlikely to care. Conversely, among the first people to move in, what kinds of people are more likely to move in? Again, those who don't care about neighborhood interaction. After all, in the presence of a strong in-group bias, it's unlikely that the first out-group entrants will receive a particularly warm welcome. This won't matter to people who have no interest in their neighbors anyway, but would be a serious cost to someone who actually wanted to socialize with the folks next door. So both entry and exit will mean that as the neighborhood becomes more ethnically mixed, the composition of people will switch towards a greater proportion of those who aren't interested in their neighbors. Which, of course, is exactly what Putnam finds.
Understand, I'm not saying this is necessarily what's happening. For all I know, Putnam's explanation - of the change in the ethnic mix causing people to change their behavior - may be the correct one. I'm only saying that the scenario I outline above is a plausible alternate explanation for his findings, and that Putnam can't dismiss a selection bias as easily as he claims to.
Of course, from a cross-sectional social capital perspective, it doesn't matter which explanation you believe. Either way it'll be true that more ethnically diverse neighborhoods will have lower social capital, and all the potential issues associated with that - if you're trying to decide what kind of a neighborhood to live in, the correlation is all that matters. But it does change the policy implications of the findings fairly drastically. If growing ethnic diversity is causing people to cut back on civic engagement, then we need to ask some hard questions on what the costs of immigration / ethnic heterogeneity are . But if it's simply the selection effect at work, then those questions become unimportant - the overall civic engagement of society isn't shrinking, it's simply being redistributed.
 Putnam, to his credit, is quick to warn against any such policy stance based on his findings. He argues that the 'hunkering down' is merely a middle-term phenomenon, and that "In the medium to long run, on the other hand, successful immigrant societies create new forms of social solidarity and dampen the negative effects of diversity by constructing new, more encompassing identities. Thus, the central challenge for modern, diversifying societies is to create a new, broader sense of ‘we’." Brave words, those, and full marks to him for saying them, but I can't help thinking that they sound too much like the words of someone who doesn't like the normative implications of his findings. Putnam is making a strong empirical claim and then tacking on what is, in effect, a prayer at the end.