In my calculus course we extended the definition of riemann integrable to functions whose domain are jordan-measurable sets, but can we extend the definition if we just ask for the domain to be bounded? like this :
Let $f:D\subset \mathbb R^n\to \mathbb R$ be a bounded function over $D$ (which is a bounded set). $f$ is said to be riemann integrable over $D$ iff the function $f_D=\begin{cases} f(x), & \text{if x$\in$ D} \\[2ex] 0, & \text{if x$\notin$ D} \end{cases}$ is riemann integrable over some rectangle $R$ such that $D\subseteq R$. In that case we define the integral of $f$ over $D$ like $\int_{D}f=\int_Rf_D\,.$
Or are there an problems if we ask $D$ just to be bounded?
Yes, you can define it like this. $D$ may not be Jordan measurable, but $f_D$ may be Riemann integrable. One such example would be $f(x)=0$ for all $x\in D$.