Well, you can always use the Taylor's series form for the exponential:

.

In particular, if H is diagonalizable, that is, if H= PDP^{-1} where D is diagonal and P some invertible matrix, (D will have the eigenvalues of H on its diagonal, and P is the matrix with the eigenvectors of H as columns) then

where "cos(D)" is the diagonal matrix having on the diagona and "sin(D)" is the diagonal matrix having on the diagonal- being the eigenvalues of H.