The closest to Visio in Linux is OpenOffice Draw. This is, going by the convention that a diagram tool is an integral part of an office or productivity suite.
I have myself done two comprehensive SRS (System Requirement Specification) documents – about 50 pages in length and full of UI diagrams – using OpenOffice Draw. The connectors, the basic shapes and numerous the export options (or file formats) are all indispensable. But what was missing was a clipart of common shapes and elements.