Given, $2x - \frac{2}{x} = 3$. Dividing by 2, we get $x - \frac{1}{x} = \frac{3}{2}$. The expression is $8\left(x^3 - \frac{1}{x^3}\right)$. $$ \begin{aligned} x^3 - \frac{1}{x^3} &= \left(x - \frac{1}{x}\right)^3 + 3\left(x - \frac{1}{x}\right) \\ &= \left(\frac{3}{2}\right)^3 + 3\left(\frac{3}{2}\right) \\ &= \frac{27}{8} + \frac{9}{2} \\ &= \frac{27 + 36}{8} = \frac{63}{8} \\ \therefore 8\left(x^3 - \frac{1}{x^3}\right) &= 8 \times \frac{63}{8} = 63 \end{aligned} $$ Thus, the value is $63$.