Asymmetric particles are way more tricky than symmetric ones, I would strongly advise you to use the algorithm with caution and really think carefully about what you want to achieve. However, I am not sure what you mean by asymmetric here, or at least not what level of asymmetry you want to achieve. We have used the algorithm before to study triangleshttp://dx.doi.org/10.1063/1.4963804https://arxiv.org/abs/1606.00223
This works well. However, something like an L-shape, which would follow a circular trajectory, will require much more thought and modification of either the algorithm (in the case of a Langevin thermostat), or clever use of building blocks with motors and non-self-propelled bits using the virtual sites. In the case of the LB simulations, I'm not presently sure how to best go about this, and I would advise you not to use the method to study that kind of shapes, although something with a mirror symmetry plane (triangle, rod, or cone) would be fine.
You can have the particle move both towards the tip and the base of the triangle, as you can see in the paper, but this requires you to properly set up the quaternion for the direction of motion, relative to the shape of your asymmetric object. Just give it a try and visualize your result, it should become really clear what the algorithm does from just looking at a few movies of the simulations. I hope this helps.