How to construct a simple complex torus of dimension $\geq 2$?

I haven't tried this by looking at conditions on the complex structure, but there is an outline in Shafarevich's book on how to do this by finding a condition on the lattice $\Lambda$.

Basically, let $\Lambda = s_1 \mathbb Z \oplus s_2 \mathbb Z \oplus s_3 \mathbb Z \oplus s_4 \mathbb Z$ where $s_i$ are vectors in $\mathbb C^2$ (the details for $n \geq 3$ are similar). Also write $s_i = (\alpha_i,\beta_i)$ for some complex numbers $\alpha_i$ and $\beta_i$.

Let $X = \mathbb C^2 / \Lambda$ be our torus, and suppose that it admits a curve $C$ (not necessarily nonsingular). The curve $C$ defines an entire 2-homology class $[C]$ by integration, and this class is non-zero (integrate a volume form on $X$ over it).

Now, as you surely know a 2-torus is diffeomorphic to $\mathbb R^4 / \mathbb Z^4$, so the 2-homology of $X$ is generated by six classes $S_{ij}$ corresponding to the 2-real-dimensional faces of $\mathbb R^4 / \mathbb Z^4$.

Consider the holomorphic 2-form $\mu = dz \wedge dw$ on $X$, and write $[C] = \sum a_{ij} S_{ij}$ for some integers $a_{ij}$. As $C$ is a curve, it carries no non-zero 2-forms, so $\mu|_C = 0$. Then

$$ 0 = \int\limits_C \mu = \sum a_{ij}(\alpha_i \beta_j - \alpha_j\beta_i), $$

by explicit calculations. You just have to calculate the integral of $\mu$ over $S_{ij}$, this gives the $\alpha_i \beta_j - \alpha_j\beta_i$ terms - the rest is by linearity. (I may be off by a factor of 4, but it doesn't matter).

So, if $X$ admits a curve, then the coefficients of the parameters $s_i$ of the lattice are linearly dependent over $\mathbb Z$. By picking some square roots of prime numbers as parameters, you can find an explicit lattice which does not satisfy this condition. The corresponding torus will thus not admit any complex curve.