It's 10F out before windchill, and I have no heat, exactly how do I "clean" my furnace?

Last winter my furnace started to do this ticking thing, where it wouldn’t ignite. (I assume.) I had a repairman come twice, and he said that it needed to be cleaned out. He said it was collecting all of the dust from the cellar and it was stuffing up the furnace. He told my husband to get in there with the shopvac and clean it out when it happens.
Well, my husband is at work for another 2 hours, and I’m sitting here freezing my butt off. I haven’t had heat for about an hour now, maybe a little bit longer. I can’t figure out what I’m supposed to clean in there. It looks clean to me.
He’d like me to wait until he can take a look at it before we call the repairman again, assuming it is the same problem we had last year. In the meanwhile.. I’m cold. I would like to take a look at it myself. Are there removable parts in there that I should be taking out to vacuum around?