I am trying to compute $$\int d^Dq \; e^{iq\cdot\left(r-r'\right)}\cos\left(c_L\left|t-t'\right|\left|q\right|\right),$$ that is, the $D$-dimensional inverse Fourier transform of $\cos\left(a\left|q \right|\right),\;a>0$. I am interested in $D=1,2,3$ and am unskilled with distributions. I have computed the $D=1$ case to be $$\pi\delta\left(\left|r-r'\right|-c_L\left|t-t'\right|\right)+\pi\delta\left(\left|r-r'\right|+c_L\left|t-t'\right|\right).$$
Edit: When transformed into spherical coordinates, the integrals become:
$D=1:$ $$2\int_0^{\infty}dq\;\cos\left(\left|r-r'\right|q\right)\cos\left(c_L\left|t-t'\right|q\right)$$ $D=2:$ $$2\pi\int_0^{\infty}dq\;q\,J_0\left(\left|r-r'\right|q\right)\cos\left(c_L\left|t-t'\right|q\right)$$ $D=3:$ $$4\pi\int_0^{\infty}dq\;q\,\sin\left(\left|r-r'\right|q\right)\cos\left(c_L\left|t-t'\right|q\right)$$
Spherical coordinates are good candidates to evaluate these integrals. The $d$-dimensional measure is given by $d^d k = k^{d-1}d^{d-1}\Omega\,dk$, where $d^{d-1}\Omega$ is the surface measure on the sphere $S^{d-1}$ and $k=|k|$. For example, in 3 dimensions, the integral would be $$\int_0^\infty k^2 dk \int_0^{2\pi}d\phi \int_0^\pi \sin \theta \,d\theta \,e^{ik|r-r'|\cos\theta} \cos(c_L|t-t'|k),$$ where I have chosen the $z$-axis as parallel to the $\vec{r}-\vec{r}'$ vector so that $\vec{k} \cdot (\vec{r}-\vec{r}')= k|r-r'|\cos\theta$.
Edit: For $d=3$, your integral is $$-2\pi\frac{d}{d|r-r'|}\int_{-\infty}^\infty \cos\left(|r-r'|q\right)\cos\left(c_L|t-t'|q\right)\,dq.$$ This is formally the same integral as for the case $d=1$, with an additional derivative. In the final result, you will have $\delta'$, the distributional derivative of $\delta$, defined by $\langle-\delta', f\rangle = \langle\delta, f'\rangle$, where $\langle g, h\rangle$ denotes the inner product $\int_{-\infty}^\infty g(x)\,h(x)\,dx$, and $f\in C^\infty_c$ is a smooth function with compact support.