Sadly, as with all things, there will be cases where dogs become sick or even die after microchipping. Some of this will be caused by microchipping and others will just be a coincidence. It's a bit like vaccination- some people will see the list of possible side effects or go off of the horror stories they hear and decide against vaccinating their children/pets.
Personally I would much rather microchip than not microchip, and have had all my animals chipped excluding exotics and fish. Several months ago my sister brought home a loose dog- he had a collar but no tag. We took him to the vets and they found he had a chip, so he was reunited with his owners. He had apparently had a tag on the collar but he must have lost it, and was found wandering a good hour's drive from where he had escaped the owner's garden. If he hadn't been chipped, it's likely he would have never found his way home.
I'm sure that there are some implanters out there that do bad jobs and of course there will be animals that react badly, as with absolutely anything (medication, vaccination, flea/worm treatment, food).
However the majority of the studies done on the correlation of microchips and tumours are done on rats/mice (as well as other bizarre animals such as shrews, bats and even a common marmoset). Not only can you not generalise findings between species, but some of these studies used genetically altered rats/mice that are extremely prone to tumours. So, until there are reliable, long-term studies done on the general dog population that can highlight any correlation between tumours and microchips, I will continue to abide by the law and microchip my dogs.