We consider the space BVA(Omega) of functions of bounded A-variation. For a given first-order linear homogeneous differential operator with constant coefficients A, this is the space of L-1-functions u : Omega -> R-N such that the distributional differential expression Au is a finite (vectorial) Radon measure. We show that for Lipschitz domains Omega subset of R-n, BVA(Omega)-functions have an L-1(partial derivative Omega)-trace if and only if A is C-elliptic (or, equivalently, if the kernel of A is finite-dimensional). The existence of an L-1(partial derivative Omega)-trace was previously only known for the special cases that Au coincides either with the full or the symmetric gradient of the function u (and hence covered the special cases BV or BD). As a main novelty, we do not use the fundamental theorem of calculus to construct the trace operator (an approach which is only available in the BV- and BD-settings) but rather compare projections onto the nullspace of A as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on Au.