It very much depends on what notes you accentuate. It's a psychological thing too. While you could play Cmajor over a Dminor chord, it would sound strange because psychologically you are playing Cmajor......yeah it's hard to explain.

Heres an AWESOME trick. you will be amazed at how cool this sounds.
Over a Dminor chord, play the Eminor7 arpeggio

--5------------------------------------7---10-
--6-------------------------------8-----------
--5------------------------7---9--------------
--7--------------------9----------------------
--5------------7--10--------------------------
-----------------------------------------------
Dminor7

It sounds awesomely jazzy because the notes you are hitting are all the notes that make the D dorian sound distinctive (D E G B) the 1st, 6th, 9th and 11th. I use this trick with my Jazz solos all the time. try it out