CRISPR-Cas is an RNA-guided adaptive immune system that protects bacteria and archaea from invading nucleic acids. Type III systems (Cmr, Csm) have been shown to cleave RNA targets in vitro and some are capable of transcription-dependent DNA targeting. The crenarchaeon Sulfolobus solfataricus has two divergent subtypes of the type III system (Sso-IIID and a Cmr7-containing variant of Sso-IIIB). Here, we report that both the Sso-IIID and Sso-IIIB complexes cleave cognate RNA targets with a ruler mechanism and 6 or 12 nt spacing that relates to the organization of the Cas7 backbone. This backbone-mediated cleavage activity thus appears universal for the type III systems. The Sso-IIIB complex is also known to possess a distinct ‘UA’ cleavage mode. The predominant activity observed in vitro depends on the relative molar concentration of protein and target RNA. The Sso-IIID complex can cleave plasmid DNA targets in vitro, generating linear DNA products with an activity that is dependent on both the cyclase and HD nuclease domains of the Cas10 subunit, suggesting a role for both nuclease active sites in the degradation of double-stranded DNA targets.