It should be possible to just simply re-save (File ► Save as..) VSD files to VSDX format in the latest version of Microsoft Visio, but is not possible using older versions of the program that did not support .vsdx format.
This seems to be the most convenient and perhaps also only reliable way how to achieve this conversion. However, LibreOffice apparently also supports at least some versions of Visio formats.
Updated: August 22, 2022