We establish upper and lower bounds on the number of connected components of lines tangent to four triangles in $\mathbb{R}^3$. We show that four triangles in $\mathbb{R}^3$ may admit at least 88 tangent lines, and at most 216 isolated tangent lines, or an infinity (this may happen if the lines supporting the sides of the triangles are not in general position). In the latter case, the tangent lines may form up to 216 connected components, at most 54 of which can be infinite. The bounds are likely to be too large, but we can strengthen them with additional hypotheses: for instance, if no four lines, each supporting an edge of a different triangle, lie on a common ruled quadric (possibly degenerate to a plane), then the number of tangents is always finite and at most 162; if the four triangles are disjoint, then this number is at most 210; and if both conditions are true, then the number of tangents is at most 156 (the lower bound 88 still applies).