Heat-killed bacteria in the Griffiths experiment would have released their chromosomes into the environment, which could then have been taken up by non-virulent bacteria. Using the new DNA, these would have become virulent by getting the genes for the smooth coat.
Work Step by Step
First consider means of DNA transfer: one of them allows for transfer from the environment. This could explain transfer of virulence to non-virulent bacteria without living virulent cells being involved.