You have the following additional information about the pieces: The Queen is an all-attacking piece, i.e. it can attack any other piece (King or Rook) from its position. On the other hand, no King can attack any other chess-piece. Rooks are friendly and won't attack each other.
Toby Gottfried created this applet to help solve the puzzle: http://www.gottfriedville.net/puzzles/grid/chess.htm
Source: Sudipta Das.
The added time and Toby Gottfried's applet led to many more solution submissions for this puzzle. With that quantity, I must conclude there is only one solution (other than its four rotations):
Q:d5, R:a5,e1, K:c3,c7,g3,g7 - - - - - - - - - - K - - - K - - - - - - - - - R - - - Q - - - - - - - - - - - - - K - - - K - - - - - - - - - - - - - R - - -A few solvers found solutions where none of the pieces were attacked. One such is:
R _ _ _ _ _ _ _ _ _ _ _ R _ _ _ _ Q _ _ _ _ _ _ _ _ _ K _ _ K _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ K _ _ K _ _ _ _ _ _ _ _ _