First integrate over $\theta_1,\theta_2$. Use the delta function representation (for $k\in\mathbb{R}$)
$$\int_{-\infty}^\infty e^{2\pi i k\theta}\,d\theta=\delta(k),$$
to evaluate 
$$\int_{-\infty}^\infty \int_{-\infty}^\infty e^{2\pi i (v_1\cdot x)\theta_1+2\pi i(v_2\cdot x)\theta_2}\,d\theta_1 d\theta_2=\delta(v_1\cdot x)\delta(v_2\cdot x).$$

Next the integral over the vector $x$. [work in progress]