  We introduce a twisted version of $K$-theory with coefficients in a
$C^*$-algebra $A$, where the twist is given by a new kind of gerbe, which we
call Morita bundle gerbe. We use the description of twisted $K$-theory in the
torsion case by bundle gerbe modules as a guideline for our noncommutative
generalization. As it turns out, there is an analogue of the Dixmier-Douady
class living in a nonabelian cohomology set and we give a description of the
latter via stable equivalence classes of our gerbes. We also define the
analogue of torsion elements inside this set and extend the description of
twisted $K$-theory in terms of modules over these gerbes. In case $A$ is the
infinite Cuntz algebra, this may lead to an interpretation of higher twists for
$K$-theory.
