Certainly the fact that the ring of differential operators is non-Noetherian is an inconvenience but it is not clear if it is more than that. For instance one can define the notion of holonomic module. It is not a direct translation of the characteristic zero definition (and this is certainly related to this inconvenience) but once given it seems to work as well as in characteristic zero:

[MR1918185](https://mathscinet.ams.org/mathscinet-getitem?mr=1918185) (2003h:14030)
Bögvad, Rikard(S-STOC)
An analogue of holonomic D-modules on smooth varieties in positive characteristics. (English summary)
The Roos Festschrift volume, 1.
Homology Homotopy Appl. 4 (2002), no. 2, part 1, 83–116.
14F10 (16S32 32C38)