What are the FAA regulations on grass airstrips?
I went poking around the FAA Regulatory and Guidance Library until I found FAR 139.307, which states:


  Sec. 139.307
  
  Unpaved areas.
  
  (a) In a manner authorized by the Administrator, each certificate
  holder [must] maintain and promptly repair the surface of each gravel,
  turf, or other unpaved runway, taxiway, or loading ramp and parking
  area on the airport that is available for air carrier use as follows:
  
  ...
  
  (b) FAA Advisory Circulars contain methods and procedures for the maintenance and configuration of unpaved areas that are acceptable to the Administrator. 


So I went looking for those advisory circulars.  There are probably more that I haven't found, but AC 150/5300-13A - Airport Design beginning at page 82 seems to have the sort of thing you're looking for.