In this paper we study continuous time filtering for linear multidimensional systems driven by fractional Brownian motion processes. We present the derivation of the optimum linear filter equations which involve a pair of functional-differential equations giving the optimum error covariance (matrix-valued) functions and the optimum filter. These equations are the appropriate substitutes of the matrix-Riccati differential equation arising in classical Kalman filtering. However, the optimum filter has the classical appearance, and, as usual, it is driven by the increments of the observed process.