Investigations (by the authors and others) of finite-size effects in systems undergoing phase transitions, and more particularly in the study of the temperature dependence of various thermodynamic properties of a spherical model of ferromagnetism confined to a restricted geometry, lead to a consideration of the lattice sums \[ {\mathcal K}_{\underline\tau}(\nu \mid m;y):=\mathop {\sum}_{\mathbf q}'\cos(2 \pi {\mathbf q} \cdot\underline\tau)(yq)^{-\nu}K_ \nu(2yq), \] where \({\mathbf q}:=(q_ 1,\dots,q_ m) \in \mathbb{Z}^ m\), \(q:=\| {\mathbf q} \|: =(\sum^ m_ 1 q^ 2_ j )^{1/2}\), \(\tau:=(\tau_ 1,\dots,\underline\tau_ m) \in \mathbb{R}^ m\) with \(0 \leq \tau_ j \leq {1 \over 2}\) \((j=1, \dots,m)\), \(y\) is a real parameter, \(K_ \nu(\cdot)\) are modified Bessel functions, and \(\sum'\) runs over all integral components of \({\mathbf q}\) with the exception of \({\mathbf q}= \text{ \textbf{0} }\). The cases \(\tau=\mathbf{0}\) (previously considered by the authors) and \(\underline\tau=(1/2,\dots,1/2)\) have particular physical significance. The sum \({\mathcal K}_{\underline\tau}\) is evaluated (making use of the Poisson summation formula) in terms of simplified summations and constants \(D_{\underline\tau}(\nu \mid m)\) which are explicitly evaluated for selected \(\underline\tau\) and \(m\), and for which otherwise some selected numerical evaluations are given. It is necessary to consider the summations with \(\nu<1\) \((\nu \neq 0)\), \(\nu>1\), separately and (as might be expected with Bessel functions) some simplifications ensue if either \(\nu\) is an integer, or half an odd integer.