We do. From here:
...Petroleum product exports are on track for another banner year, with total exports climbing to a 3.7 million barrels per day (bbpd) average for 2014... ...Overall, the U.S. remains a net importer of crude oil and refined petroleum products. But, data from the EIA show that reliance on oil from other countries has been declining over the past 8 years, from its peak in August 2006.
We can export condensates and refined products but not crude by law.