A method of making an embossed air-laid absorbent sheet product includes depositing a dry web of fibers on a foraminous support; compacting the fiber web; embossing the compacted web with a laser-engraved roll with a hard rubber embossing surface, the hard rubber embossing surface having a plurality of sinuate bosses radially projecting circumferentially around the laser-engraved roll to provide the fiber web with a functional emboss pattern including a plurality of continuous sinuate compressed regions extending in a machine direction of the fiber web, the laser-engraved roll being heated to or maintained at a temperature of from about 100° F. to about 300° F.; and bonding the fiber web.