We develop an adaptive finite element method (AFEM) using piecewise linears on a sequence of triangulations obtained by adaptive √3 refinement. The motivation to consider √3 refinement stems from the fact that it is a slower topological refinement than the usual red or red-green refinement, and that it alternates the orientation of the refined triangles, such that certain features or singularities that are not aligned with the initial triangulation might be detected more quickly. On the other hand, the use of √3 refinement introduces the additional difficulty that the corresponding finite element spaces are nonnested. This makes the setting nonconforming. First we derive a BPX-type preconditioner for piecewise linears on the adaptively refined triangulations and we show that it gives rise to uniformly bounded condition numbers, so that we can solve the linear systems arising from the AFEM in an efficient way. Then we introduce the AFEM of Morin, Nochetto, and Siebert adapted to our special case for solving the Poisson equation. We prove that this adaptive strategy converges to the solution within any prescribed error tolerance in a finite number of steps. Finally we present some numerical experiments that show the optimality of both the BPX preconditioner and the AFEM.