The "guts" of a sound absorption panel is the material inside that reduces high- to mid-frequency sound by physically impeding the movement of sound in air. The covering of an absorption panel must allow sound to pass through to some extent. Unlike speaker coverings, which should be as acoustically transparent as possible, it is OK if the cover absorbs some sound. Burlap, or rough-textured fabrics of large fibers with space between them, is ideal. The texture of the material starts to break up higher frequencies even before sound hits the absorbing material.
Muslin is a light cotton fabric of loose weave, and therefore works similarly to burlap in allowing sound to pass. The visual look is less "rough" than that created by burlap. Though it is usually sold in white or unbleached colors, it is easy to dye and it readily takes paint as well, so matching decor is a simple step. Though paint will reduce some of its sound transmitting properties, muslin is light enough that this will not affect frequencies that will likely be problematic.
Using wire or fiberglas window screen material is another option. Once again, sound easily passes through to the absorbent material, but where aesthetically burlap may be tricky to line up, screen material has more dimensional stability. It provides a different look that may be more suited to certain decor in dual-purpose rooms.
It is possible to take sound absorption to an extreme. High frequencies are the easiest to address due to short wavelengths, and sometimes too much high frequency information is absorbed, exaggerating the effects of mid- and low-frequency. Using ceiling tiles may be an effective solution to add mass, thus lowering the effective frequency absorption of a panel, while reflecting some high frequency back into the room. This may be helpful in improving clarity in an otherwise acoustically dead room.