The US census bureau does not answer this question directly…
According to the census (http://www.census.gov/hhes/www/housing/ahs/ahs05/tab3-15.pdf), about 33% of owner-occupied homes are owned free and clear.
However, this doesn’t include the statistics for non-owner-occupied homes. In other words, the owner of the apartment building down the street may or may not have a mortgage on the building. I think it is a good bet that a lower percentage of these income properties are owned outright, which would lower the overall percentage of units that are owned free and clear.
This type of census would also not take into account something like the foreclosure explosion because the US census figures only take into account OCCUPIED homes. An overwhelming majority of unoccupied homes are not owned free and clear.