I carry a valid US passport. I have a 30-day stay stamped at Abu Dhabi airport. After 30 days, can I leave the UAE for a few days, then come back to Abu Dhabi and get another 30-day stay visa?
Yes. The easiest / cheapest way for an American national is to cross the border by car into Oman, and return. It will restart the 30-days. You can make it your monthly trip to someplace nice and nearby like Beirut (etc.)
However, as an aside, I recommend you spend some time in Oman, whether camping or staying at the Shangri-La or wherever, it's beautiful! Then, return to get your tourist visa.
EDIT As requested: Things can always change; in 2010 Canadians could no longer do the "visa run," like they used to. This was due to a dispute about airlines that resulted in changes in their visa structure. I commented only to suggest that you check the latest news when it comes to UAE.