Yes. We will come up with $g$ maps from the Riemann surface to the two-dimensional torus. We can then take the product of these maps. As long as one of them is nonconstant at each point, it will be an embedding.

View the Riemann surface in the standard way as a handlebody that looks like a bunch of tori glued together. (Presumably the classification of Riemann surfaces, which we use to put it in this form, counts as purely topological reasoning!) There are $g$ tori. Pick one, and contract all but that one to a point. If that is one of the tori at the ends, then all the rest of the surface gets contracted to a single point. If it's in the middle, the left half gets contracted to a point on the left side of the torus and the right side gets contracted to a point on the right side of the torus.

This set of $g$ maps gives us the associated kind of map. Clearly, this gives an isomorphism on homology. To make sure this is an embedding, we have to make sure that no point on the torus gets contracted in every single map. We will do this by defining the maps a little generously, such that the stuff that is contracted for the leftmost torus ends a little bit to the right of where the stuff contracted for the second-leftmost torus begins.

Hopefully this explanation makes sense! My picture-drawing skills are not up to the task of realizing this.