Solution:

Squaring $a\sqrt{2} + b\sqrt{3} + c\sqrt{5}$ gives $$2a^2 + 3b^2 + 5c^2 + 2ab\sqrt{6} + 2ac\sqrt{10} + 2bc\sqrt{15}.$$ Matching coefficients yields $2ab = 104,$ $2ac = 468,$ $2bc = 144,$ that is $ab = 52,$ $ac = 234,$ $bc = 72,$ along with $2a^2 + 3b^2 + 5c^2 = 2006.$

Then $(abc)^2 = (ab)(ac)(bc) = 52 \cdot 234 \cdot 72 = 876096 = 936^2,$ so $abc = 936.$ As a check, $a = \frac{abc}{bc} = 13,$ $b = \frac{abc}{ac} = 4,$ $c = \frac{abc}{ab} = 18,$ and $2 \cdot 169 + 3 \cdot 16 + 5 \cdot 324 = 338 + 48 + 1620 = 2006,$ as required.

Therefore $a \cdot b \cdot c = 936.$