Not true, mostly. The states vary, but at least some have held otherwise. For example, the Arkansas Supreme Court (the most recent state court to address the issue, in 1986), found that Arkansas had to recognize out-of-state cousin marriages even though cousin marriage was expressly forbidden in Arkansas. A similar case was decided on a technicality in Arizona in 2005.
I don't think there has ever been a federal Constitutional challenge on the issue via Full Faith and Credit, so the federal courts have never considered the issue. Hard to see them finding for the state looking to deny recognition, though.
I'll try to find the case law. Part of the rationale that President Bush had for the Federal Marraige Amendment was the FF&C clause, but legal scholars pointed out at the time that no court had ever used the FF&C clause to force a state to accept any marriage contrary to its public policy.
State courts might be a different issue.
I believe this is the Arizona case, though. Seems to me like they're quite clearly leaning towards the voided category, in that they chose to apply Arizona law rather than Virginia law.
http://www.docstoc.com/docs/19666380/Cook-v-Cook-Arizona-Court-of-Appeals-1