When the taxpayers or a foundation's donors ask how their money has been spent, funding agencies need accurate information for accountability. If continued funding of research is important to you, then citing the grant is good practice. Cite the grant whether your research was directly funded or you received indirect support through services subsidized in a funded Core or Center.
Funder IDs from the Funder Registry are usually recognized "automagically" if you cite the Grant ID number. However, if there is no Grant ID number to cite, then look up your funder in the Funder Registry to credit them directly.