The diagonals of a rhombus are 6 inches and 8 inches. What is the perimeter, in inches, of the rhombus below? [asy]
import geometry; size(150); defaultpen(linewidth(0.8));
pair A = (0,0),B = (8,6),C = (16,0),D = (8,-6), E = (8,0);
draw(A--B--C--D--cycle);
draw(A--C,StickIntervalMarker(2,2,10)); draw(B--D,StickIntervalMarker(2,1,10));
draw(rightanglemark(A,E,B,30));
[/asy]
The rhombus is split into four congruent right triangles. Each right triangle has a leg of $\frac{6}{2}=3$ inches and another leg of $\frac82=4$ inches. We use the Pythagorean Theorem to solve for the length of the hypotenuse, or we recognize that $3$ and $4$ are part of the Pythagorean triple $(3,4,5)$, so the length of the hypotenuse is $5$ inches. Each side of the rhombus is $5$ inches long, so the perimeter is $4(5)=\boxed{20}$ inches.