new laundry issues

When Nate was really little, he'd go through three or four outfits a day, mostly due to leaky diapers. So we (aka Thad) were constantly doing laundry. He is no longer as fashion conscious, and fits much better in his Pampers; however, we have a new laundry conundrum. Now that Nate's eating solid foods, we have a regular stream of dirty bibs. We are going through them like hotcakes! They're all pretty small, and it would take at least a week to amass enough dirty bibs to make a load of laundry. We probably have two to three messy bibs (plus food-covered clothing sometimes too!) a day, and I don't want to put them in the basement for fear of attracting fun household creatures like mice, so they've been piling up in the kitchen (clearly a much better place for mice than the basement.) For now, the dirty ones just pile up on the counter. We've tried rinsing them, but then they're wet and we don't really know where to put them, so those get spread all over the kitchen. I'm not so excited about piles of dirty bibs, wet or dry, sitting out in 90-degree temperatures. I'm still not convinced it's better to do tons of little loads of laundry than it would be to just use something disposable, but I feel environmentally unconscious using disposable stuff when we're home and could easily use the old school version. Oh, the trials and tribulations of parenthood!