Answer: B

Solution:

Adding the three equations gives $$\begin{gathered} \left(x + \tfrac1y\right) + \left(y + \tfrac1z\right) \\ {}+ \left(z + \tfrac1x\right) \\ = 4 + 1 + \tfrac73 \\ = \tfrac{22}{3}. \end{gathered}$$

Multiplying them gives $$4 \cdot 1 \cdot \tfrac73 = \tfrac{28}{3}.$$

Expanding the product, $$\begin{aligned} &\left(x + \tfrac1y\right) \\ &\quad {}\cdot \left(y + \tfrac1z\right) \\ &\quad {}\cdot \left(z + \tfrac1x\right) \\ &= xyz \\ &\quad {}+ \left(x + y + z + \tfrac1x + \tfrac1y + \tfrac1z\right) \\ &\quad {}+ \frac{1}{xyz}. \end{aligned}$$ The middle group is the sum $\tfrac{22}{3},$ so $xyz + \dfrac{1}{xyz} = \tfrac{28}{3} - \tfrac{22}{3} = 2.$

Hence $(xyz - 1)^2 = 0,$ so $xyz = 1.$

Thus, the correct answer is B.