I guess you want the generators for the $K_0$ part (actually the generators for the $K_1$ part are interesting too, as the 4-th generator (besides the obvious 3 unitaries which generate the algebra) is somewhat non-trivial to construct.

The key idea to write down the corresponding modules is to notice that the bimodules are given as $\mathcal{S}(\mathbb{R}^p \times \mathbb{Z}^q\times F)$, but there exist different representations of the algebra on them, and distinct representations lead to distinct classes in the K-theory.

I am going describe how this works for the 2-torus, as all the interesting things happen already there, and the n-torus case can be extrapolated easily from there. The modules here are either the trivial ones ($\simeq \mathcal{S}(\mathbb{Z}^2)$ as modules), or $\mathcal{S}(\mathbb{R})$.

If $\theta$ is irrational, we have $K_0(\mathbb{T}^2_\theta) = \mathbb{Z}\oplus\theta \mathbb{Z}$, and the algebra $\mathbb{T}^2_\theta$ has cancellation (i.e. if $V\oplus W \simeq V'\oplus W$, then $V \simeq V'$ for projective modules $V,V',W$), hence we have for each $(\mathbb{Z} + \theta \mathbb{Z})^+$ a projective module that represents the class.

The projections corresponding to these classes are the Powers-Rieffel projections, as described in many places. What seems to be less well-known is how to find the corresponding projective module. This was actually written down by Rieffel in "$C^*$-algebras associated with irrational rotations" and "The cancellation theorem for projective modules over irrational rotation $C^*$-algebras".

Rieffel observed that given a strong Morita equivalence bimodule between two $C^*$-algebras $A$ and $B$, with $A$ and $B$-valued inner products $\langle , \rangle_{A}$ and $\langle , \rangle_{B}$, the element $\langle x,x\rangle_A\in A$ is a projection if and only if $ \langle x,x\rangle_B x = x$. The modules $\mathcal{S}(\mathbb{R})$ are the equivalence bimodules used for the noncommutative torus.

For all integers $p,q$, $q\neq 0$, the smooth subalgebra of the $C^*$-algebra of $\mathbb{T}^2_\theta$ acts on $\mathcal{S}(\mathbb{R})$ on the right, with action given by $f(t)\triangleleft U_1 = f(t-pq+q\theta)$ and $f(t)\triangleleft U_2 = e^{2\pi i t/q} f(t)$, for the generators $U_1$ and $U_2$ of the noncommutative $2$-torus. If $a,b\in\mathbb{Z}$ are such that $aq-pb=\pm 1$, there is a left action of the algebra $\mathbb{T}^2_{(a\theta +b)/(p+q\theta)}$, which commutes with this right action.

A Powers-Rieffel projection can be constructed from such a module, for example, using a function $f$ which is strictly positive on the interval $[0,\theta]$ and supported on some interval with length less than $1$, and then appropriately normalizing this function, so that it's inner product becomes $1_B$.

The trace of $\langle f,f\rangle_A$ then becomes(as described in Theorem 1.4 of the cancellation theorem paper) $|q \theta + p|$. If we take an extra finite group $\mathbb{Z}/d\mathbb{Z}$ in the bimodule, the trace becomes $d|q \theta + p|$. Hence the modules $\mathcal{S}(\mathbb{R}\times F)$, with the appropriate action, give you the appropriate classes of $K_0$.

For higher-dimensions, we have more Powers-Rieffel projections, coming from choosing different actions on the various $\mathcal{S}(\mathbb{R}^p\times \mathbb{Z}^q \times F)$.

**EDIT** I should add that unfortunately, I was too optimistic about the actual computation of the actual projections. If $p=1, q=n-2$, the computation can be done (and leads to projections with trace $\theta_{ij}$), however, I am unable to compute explicit projections coming from modules with $p>1$, and various discussions with other people in the field lead me to believe this is actually an open problem.