Sure, that kind of modal shifting using the same root relationships is generally quite effective. In your particular example, you might want to raise the 7th scale degree (C to C#) for V and vii chords (much as you might do in minor) in order to get a stronger drive to the i chord, but then you wouldn't strictly be in D Dorian. Either way works but has a rather different character.
A lot of music in Dorian doesn't use V as often to lead to I as in Major. VII - I (in your D Dorian example, C major to D minor) is quite common and effective.
The only mode in which I - IV - V - I root relationships tend to not "work" quite as well is Locrian. The diminished fifth above the tonic tends to throw things off. For example, in B Locrian (BCDEFGAB), the I chord would be a diminished triad, and that tends to weaken its use as a tonic. Most composers will raise the fifth scale degree so that the I chord is just minor. Also, in both Locrian and Phrygian, the II chord is used very often to lead to I. For example, in E Phrygian (EFGABCDE), F major moving down to E minor.
So to sum up, yes, the same sort of root relationships often do "work" in a similar though complementary fashion in the other church modes, though sometimes a scale degree is altered for some chords. However, new progressions specific to these other modes, such as bVII - I in Mixolydian and Dorian and Aeolian and bII - I in Phrygian and Locrian, also sometimes start getting used in their place (or in addition).