So, just some terminology to get out of the way at the beginning: archaeologists generally refer "Anasazi" cultures as "Ancestral Puebloan" cultures these day, and so I'll use that term for the rest of this answer.

So, to begin with, a very great number of Native groups in the American Southwest claim some sort of connection to Ancestral Puebloan archaeological sites, especially those in the Four Corners region (including Mesa Verde National Monument) and in Chaco Canyon, NM. These areas are seen as the origin point for most modern Pueblo groups and feature prominently in their historical accounts.

Most importantly, it would be mistaken to talk about these sites as "abandoned" since that is very much not how modern Puebloan people (and Puebloan people historically) understand these sites. Modern Puebloan people generally don't speak of these locations as "abandoned" because they view them as still being the homes of their ancestors. To understand this we have to abandon the very capital-M Modernist idea of linear history that posits a gulf between past and present.

It's not that Puebloan people don't have a sense of past and present - it is understood that the people who lived at these sites lived their in the past - but in Puebloan thought, just because something happened in the past doesn't make it inaccessible to those in the present. These "abandoned" sites are seen as living places despite having fallen to ruin and that those who lived there are no longer alive. This concept can be understood in terms of ownership, that those sites still belong to the people who built them and lived there. This short (9:30) video produced by the BLM and a number of other organizations gives a very good overview of how modern Pueblo groups conceive of these ancestral sites, using their own words.

The other angle of this is that these sites and regions generally feature prominently in the origin stories of most modern Pueblo groups. These locations are very explicitly called out as places of cultural origin for the modern groups.

For both these reasons, these Ancestral Puebloan sites are often locations of ceremonies and other rituals conducted by modern Puebloan people.

Between this understanding of the sites as still living and their position as cultural origin points, these sites never really left the cultural understanding of Pueblo groups. Because of that, they consequently never entered into a category of "historical" that Europeans (and Americans) coming from a Modernist perspective would label archaeological sites with. Without that view of these sites as "historical" or "archaeological" there isn't necessarily a cultural logic that would subject them to excavation or archaeological research. The large presence of these sites in oral histories also further obviates some of the motivation for archaeological work, in the modern sense, as a way to understand periods that have passed out of history, whether written or oral.

I should also mention, at least briefly, that some non-Pueblo groups also feature these sites in their oral histories. In particular, the Navajo claim these Ancestral Puebloan sites as part of their cultural heritage. Probably one of the most famous connections to these Ancestral Puebloan sites is the Navajo story of the Gambler. The Gambler is a nefarious figure who comes to Chaco Canyon and engages the people there (in the story, ancestral to the Navajo and the Pueblos) in various wagers which he always wins. The people, having lost all their goods, wager their freedom and again lose to the Gambler who then forces them to build the large structures we see in the Canyon today. The people eventually manage to free themselves from the Gambler and disperse from Chaco Canyon to their current locations. You can read one particular account (in part) of this story here.