Early Christians refused to do this. They were accused of being "atheists", because they didn't at least acknowledge the official Roman gods. The persecutions were more severe than what many anti-Christian apologists claim they were, but the intensity of the persecution varied from Emperor to Emperor, and with the various Governors of the provinces.
The Christians ignored the Roman gods but got into trouble because they refused to offer sacrifices to “god Caesar”.
The same as we today are condemned because we refuse to offer sacrifices to “god Obama”.