The volume of a certain rectangular solid is $216\text{ cm}^3$, its total surface area is $288\text{ cm}^2$, and its three dimensions are in geometric progression.  Find the sum of the lengths in cm of all the edges of this solid.
1. Let the dimensions of the rectangular solid be \(a/r\), \(a\), and \(ar\), where \(a\) is the middle term and \(r\) is the common ratio of the geometric progression.
2. The volume of the rectangular solid is given by:
   \[
   \frac{a}{r} \cdot a \cdot ar = a^3 = 216 \text{ cm}^3
   \]
   Solving for \(a\):
   \[
   a^3 = 216 \implies a = \sqrt[3]{216} = 6
   \]
3. The total surface area of the rectangular solid is given by:
   \[
   2\left(\frac{a^2}{r} + a^2 + a^2 r\right) = 288 \text{ cm}^2
   \]
   Substituting \(a = 6\):
   \[
   2\left(\frac{6^2}{r} + 6^2 + 6^2 r\right) = 288
   \]
   Simplifying:
   \[
   2\left(\frac{36}{r} + 36 + 36r\right) = 288
   \]
   \[
   \frac{36}{r} + 36 + 36r = 144
   \]
   Dividing by 36:
   \[
   \frac{1}{r} + 1 + r = 4
   \]
   Let \(x = r + \frac{1}{r}\):
   \[
   x + 1 = 4 \implies x = 3
   \]
   Solving the quadratic equation \(r + \frac{1}{r} = 3\):
   \[
   r^2 - 3r + 1 = 0
   \]
   Using the quadratic formula:
   \[
   r = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2}
   \]
4. The sum of the lengths of all the edges of the rectangular solid is:
   \[
   4\left(\frac{a}{r} + a + ar\right)
   \]
   Substituting \(a = 6\):
   \[
   4\left(\frac{6}{r} + 6 + 6r\right)
   \]
   Using \(r = \frac{3 + \sqrt{5}}{2}\) and \(r = \frac{3 - \sqrt{5}}{2}\), we find:
   \[
   \frac{6}{r} + 6 + 6r = \frac{6}{\frac{3 + \sqrt{5}}{2}} + 6 + 6 \cdot \frac{3 + \sqrt{5}}{2}
   \]
   Simplifying:
   \[
   \frac{6 \cdot 2}{3 + \sqrt{5}} + 6 + 3(3 + \sqrt{5})
   \]
   \[
   \frac{12}{3 + \sqrt{5}} + 6 + 9 + 3\sqrt{5}
   \]
   Rationalizing the denominator:
   \[
   \frac{12(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})} + 15 + 3\sqrt{5}
   \]
   \[
   \frac{12(3 - \sqrt{5})}{9 - 5} + 15 + 3\sqrt{5}
   \]
   \[
   \frac{12(3 - \sqrt{5})}{4} + 15 + 3\sqrt{5}
   \]
   \[
   3(3 - \sqrt{5}) + 15 + 3\sqrt{5}
   \]
   \[
   9 - 3\sqrt{5} + 15 + 3\sqrt{5} = 24
   \]
   Therefore:
   \[
   4 \times 24 = 96
   \]

The final answer is \(\boxed{96}\).