On the $d$ - dimensional torus, for every irrational point $\theta = (\theta_1,...,\theta_d)$ (i.e $\theta_i \notin \mathbb{Q}$), the orbits of $SL_d(\mathbb{Z})$ on this point are dense.
However, there is something bothering me - obviously if $\theta_i \in \mathbb{Q}$ for every $1\leq i\leq d$ the orbit is not dense, and furthermore, it is finite (we can just take a common divisor between all the $\theta_i$'s). However, when there are irrational and irrational $\theta_i$'s, I dont think the orbit can be finite (for example, for $(1/2, 1/\sqrt{5})$ on $\mathbb{T}^2$ - Im pretty sure the orbit being finite contradicts the fact $1/\sqrt{5}$ is irrational).
Maybe badly approximable numbers cannot have dense orbits on such points? Sounds a little bit like an overkill, but this is the best I got right now.