The majority of my Arkansas civil war ancestors are buried at the place they lived. Their graves don't have headstones. I suggest that you find your ancestor in census records 1860 and 1870(if still living by 1870); get a platt map of that county & find the homesite. If marked, most would have been with a rock marker, possibly inscribed, or with a cairn(pile of rocks) at that time. Granite markers would have been too expensive for most working class people at that time. Also, be ready for you not being able to actually find the gravesite. It is a possibility that there are no clues to the burial site due to the landscape changes between then & now. It's been my experience that my ancestors who lived during this time period were given no markers until later generations placed a stone on their grave. Even if you find the grave site, be ready for the property owners not granting you access. It will be extremely costly for you to "prove" that there is a grave there without actually disturbing the grave. Arkansas laws do allow descendants access to gravesites on private property, but these are gravesites that are already established w/ markers.