The first thing that comes to mind is that the next chord should be Am or F, in which case talking about C and E being chromatic mediants feels a bit too fancy. For talk about chromatic mediants to make sense, shouldn't you define some context - where's your tonic, and what do you do with the chords? Other interpretations are that the assumed tonic is an E based chord, either E major as in Dekkadeci's answer, or E minor that's exchanged for a major as in Albrecht Hügli's answer.
I'm sure there are more interpretations, but let's sum up these ones:
- (1) Am as tonic : the C - D - E chords are III - IV - V
- (2) E as tonic : the C - D - E chords are ♭VI - ♭VII - I
- (3) expected Em as tonic : the C - D - E chords are VI - VII - I (major)
The "chromatic mediant with a passing chord" explanation feels a bit far-fetched, IMO, when simpler explanations are available. :) Or maybe the interpretation (3) above could be seen as equivalent, depending on how the chords are stressed.