There are several drive types in various games and shows that avoid dilation by not actually generating velocity.
The 2300AD game's stutterwarp involves macro-scale quantum tunneling effect to generate both high sublight through several hundred C.
Star Trek makes use of "Heisenberg Compensators" that negate dilation effects at both high sub-C and at warp. Further, at warp it "moves a bubble of space" rather than the ship itself through space, which theoretically could avoid some or all dilation effects as well, and there are implications in some novels that warp and impulse are related effects. (Some of those may be unintentionally making use of Amarillo Design Bureau's Star Fleet Universe, a spin-off setting from TOS/TAS for their game lines, originally a sublicense from Franz Joseph. in the SFU, impulse drives can generate warp fields and do low warp speeds.)
The Spelljammer setting for AD&D (and its derived stories) simply redefines the universe to ignore quantum physics.
The Novels (and the related GURPS sourcebook) for the Lensmen setting make use of a drive which disconnects the ship from accumulating inertia. Doc Smith seems to have believed that Inertia and Dilation were in fact closely linked, and that the Inertialess ships would be limited in speed only by friction. While most consider the Inertialess Drive to be FTL, it operates sublight as well.
Many harder settings also ignore relativity - some by predating common awareness of it, a few by predating it (Verne's works).
The general pattern is to simply keep sublight speeds under 0.8C where it isn't really noticed.
As a corollary, almost any good means of evading dilation also enables FTL, since the inability to attain C is a function of dilation.