  We consider a class of geometrically frustrated Heisenberg spin systems which
admit exact ground states. The systems consist of suitably coupled
antiferromagnetic spin trimers with integer spin quantum numbers $s$ and their
ground state $\Phi$ will be the product state of the local singlet ground
states of the trimers. We provide linear equations for the inter-trimer
coupling constants which are equivalent to $\Phi$ being an eigenstate of the
corresponding Heisenberg Hamiltonian and sufficient conditions for $\Phi$ being
a ground state. The classical case $s\to\infty$ can be completely analyzed. For
the quantum case we consider a couple of examples, where the critical values of
the inter-trimer couplings are numerically determined. These examples include
chains of corner sharing tetrahedra as well as certain spin tubes. $\Phi$ is
proven to be gapped in the case of trimer chains. This follows from a more
general theorem on quantum chains with product ground states.
