By means of Fourier transform microwave spectroscopy of a supersonic molecular beam, we have detected the singly substituted carbon-13 isotopic species of C(5)H, C(6)H, and C(7)H. Hyperfine structure in the rotational transitions of the lowest-energy fine structure component ((2)Pi(12) for C(5)H and C(7)H, and (2)Pi(32) for C(6)H) of each species was measured between 6 and 22 GHz, and precise rotational, centrifugal distortion, Lambda-doubling, and (13)C hyperfine coupling constants were determined. In addition, resolved hyperfine structure in the lowest rotational transition (J = 32-->12) of the three (13)C isotopic species of C(3)H was measured by the same technique. By combining the centimeter-wave measurements here with previous millimeter-wave data, a complete set of (13)C hyperfine coupling constants were derived to high precision for each isotopic species. Experimental structures (r(0)) have been determined for C(5)H and the two longer carbon-chain radicals, and these are found to be in good agreement with the predictions of high-level coupled-cluster calculations. C(3)H, C(5)H, and C(7)H exhibit a clear alternation in the magnitude and sign of the (13)C hyperfine coupling constants along the carbon-chain backbone. Because the electron spin density is nominally zero at the central carbon atom of C(3)H, C(5)H, and C(7)H, and at alternating sets of carbon atoms of C(5)H and C(7)H, owing to spin polarization, almost all of the (13)C coupling constants at these atoms are small in magnitude and negative in sign. Spin-polarization effects are known to be important for the Fermi-contact (b(F)) term, but prior to the work here they have generally been neglected for the hyperfine terms a, c, and d.