The Collaborative International Dictionary of English v.0.48:

Grime \Grime\, n. [Cf. Dan. grim, griim, lampblack, soot, grime,
   Icel. gr[imac]ma mask, sort of hood, OD. grijmsel, grimsel,
   soot, smut, and E. grimace.]
   Foul matter; dirt, rubbed in; sullying blackness, deeply
   ingrained.
   [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Grime \Grime\, v. t.
   To sully or soil deeply; to dirt. --Shak.
   [1913 Webster]

WordNet (r) 3.0 (2006):

grime
    n 1: the state of being covered with unclean things [syn:
         dirt, filth, grime, soil, stain, grease,
         grunge]
    v 1: make soiled, filthy, or dirty; "don't soil your clothes
         when you play outside!" [syn: dirty, soil, begrime,
         grime, colly, bemire] [ant: clean, make clean]

Moby Thesaurus II by Grady Ward, 1.0:

25 Moby Thesaurus words for "grime":
   begrime, bemire, bemud, besmoke, besoil, dirt, dirty, dirty up,
   dust, filth, foul, mire, muck, muck up, mud, muddy, scum, slime,
   smirch, smoke, smooch, smudge, smut, soot, tarnish