If I play the chords: C major, D major, then E major. I'm wondering what their relationship is called.
It almost sounds like chromatic mediants but I can't call it that since it doesn't move by a third, they move by a major second.
So we just call their relationship chromatic? and that's it, or is there a special terminology?
It sounds like a chromatic mediant because C and E major are themselves chromatic mediants. You've just added in a passing chord between them.
We could call this "planing," which is just moving a particular chord shape or type up and down by parallel motion. Planing often stays within a key (it's thus called "tonal planing"), meaning that the chord qualities might change. Planing in C major, for instance, would give C, Dm, and Em. But in this case, you have what we'd call "real planing," meaning that you keep all of the relationships among each chord the same.
In a more tonal context, the chords C, D, and E can be interpreted as ♭VI-♭VII-I of E major (with substantial borrowing from the tonic minor). This interpretation can be questioned if the next chord is, say, an F major chord.
C D E can be interpreted as VI VII I in minor whereby the final chord is substituted by its relative major chord.
This is maybe the same signification or function as in Dekkadeci’s meaning. It’s surely depending of the context.
What's the relationship between the chords Cmaj Dmaj Emaj?
Here are some possible answers. The 'correct' answer is the one which is most useful in context, i.e. has the greatest explanatory power. Since you haven't given us any context, I'll leave the decision to you.
The most universal answer is that they all reduce to the same prime form and therefore all belong to the same set class: {0,3,7}. Additionally, they are all transpositionally related to each other: the progression is formed by 2 successive 2-semitone transpositions of this set class. I say "most universal" because these are basic facts--truisms even--which are always true independent of the context or particulars. But it might not be the most useful answer.
If all the voices are moving in parallel, then it is an instance of parallelism, aka planing. Planing, such as when found in impressionist music, often serves more of a melodic role than harmonic: a melody of chords. If this is the case, then I think you're asking the wrong question. Instead, it might be more useful to think about the relationships between the roots and which collection they belong to (eg: diatonic, whole tone, etc).
As has already been noted by you and Richard, C and E have a chromatic mediant relationship, and the D can be seen as a passing chord between them. IMO this answer is most convincing if the voices are not moving in parallel.
If a tonal/functional relationship is what you seek, there is no obvious answer, and certainly nothing diatonic. But there are explanations involving borrowed chords or modal mixing. Other answers have already listed some possibilities, and I bet we could brainstorm several more. bVI - bVII - I in E major and VI - VII - I in E minor (a Picardy cadence) both sound plausible.
Another tonal option is that this is part of a modulatory passage from G major to A major with D acting as a pivot chord. C to D is IV - V in G, and D to E is IV - V in A.
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:
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.
