Straining won't help with most pollutants (unless you're using RO or some serious filtration, in which case why use rain water as your source at all?), but it will get rid of insects, dust, and maybe even pollen. However, I'll agree with you that the earlier poster was probably exaggerating pretty severely. Our air and water quality in the U.S. has actually improved remarkably over the last few decades, due to a combination of tighter restrictions on motor vehicles, monitoring and restrictions on factory emissions, and the shifting of a large portion of our manufacturing overseas. There are some rivers and watersheds that were essentially toxic in the 1970s but are now pretty safe to swim and even fish in.
That being said, I'll also agree that if you live in an agricultural area, the large scale spraying of pesticides/herbicides/fertilizers will almost certainly get into your rainwater, and probably your local river/lake water as well. I don't know if it would be at levels that are high enough to effect your plants at all (probably not), but it'll be there.
If you're in the suburbs and there aren't any major industrial or agricultural areas upwind (prevailing winds, which usually means to your west in the U.S.), your rainwater is probably pretty clean. If you're right next to a major highway, or in the heart of an urban center, you're probably still getting a lot of particulates from vehicle exhaust, even if the per-vehicle emissions are better now than they were 20-40 years ago. Once again, I don't know if the pollutant levels will be high enough to effect your plants (again, probably not), but there will be measurable pollutants.
In sum, even in the worst areas I'd guess your plants won't suffer from the pollution in your rainwater. If you're in suburbia (and not downwind from an industrial park or agrobusiness district), you'll probably have pretty damn clean rainwater, almost certainly cleaner than your tap water, once you filter out the dust/pollen.