Conway and Lagarias observed that a triangular region $T_2(n)$ in a hexagonal lattice admits a {\em signed tiling} by 3-in-line polyominoes (tribones) if and only if $n\in \{3^2d-1, 3^2d\}_{d\in \mathbb{N}}$. We apply the theory of Gröbner bases over integers to show that $T_3(n)$, a three dimensional lattice tetrahedron of edge-length $n$, admits a signed tiling by tribones if and only if $ n \in \{3^3d-2, 3^3d-1, 3^3d\}_{d\in \mathbb{N}}$. More generally we study \emph{Gröbner lattice-point enumerators} of lattice polytopes and show that they are (modular) quasipolynomials in the case of $k$-in-line polyominoes. As an example of the ``unusual cancelation phenomenon'', arising only in signed tilings, we exhibit a configuration of 15 tribones in the $3$-space such that exactly one lattice point is covered by an odd number of tiles.