I planted this shrub 3 years ago and it still hasn't set flowers.
Any suggestions on how I can get it do so?

Other than normal care, I think it just needs time to mature. They seem to like moist soil, at least, mine in moist areas sucker a lot and grow faster, where when I have them in dry areas they are slow.

I wonder if lack of blooms could be from a nutrient deficiency; have you fertilized your Carolina Allspice (Calycanthus floridus) in the three years since it's been planted? Maybe late fall or early next spring you can give it a good feeding which will possibly help in production of blooms next year.

Calycanthus blooms on old wood. That means if it were to flower this year, it would have produce those flower buds the season before. The buds then survive through the winter and develop the following season. If you trimmed this shrub in the meantime, you could be trimming off flower buds.

I can't see how big the plant is, but if you haven't trimmed, and it stands about two and a half feet high (or less) now, then it is fairly normal. It will bloom next year.

Light green leaves are also normal for the plant, as long as the leaves are a solid color, and there isn't any easily seen light green and dark green areas. Calycanthus should never need fertilizing to be healthy and bloom.