Our 3-week old seems to be spitting up a lot after each breast-feeding. .. No fever, but we're wondering what nourishment, if any, he's getting?

Gaining weight? Many babies spit up some of their feedings. This may be due to overfeeding or just him gulping down air along with the milk. Burping frequently can help the latter. The most important thing is whether he is growing. Doctors use charts to plot babies' growth. If his weight and length are going along the curve, he's doing ok. http://www.cdc.gov/growthcharts/data/who/grchrt_boys_24lw_9210.pdf.
Spitting up. If he is gaining weight, growing, and having regular stools - no worry. Breastfed babies tend to overeat because they like the closeness, warmth and flavor of your milk. The valve that keeps the milk in his stomach can be a little floppy the first few months of life so when his stomach squeezes some comes back up. Keep him propped up after feeds for 40 minutes. If he is not gaining wt get him seen.